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15 Venn Diagram Questions And Practice Problems (Middle & High School): Exam Style Questions Included

Beki christian.

Venn diagram questions involve visual representations of the relationship between two or more different groups of things. Venn diagrams are first covered in elementary school and their complexity and uses progress through middle and high school.

This article will look at the types of Venn diagram questions that might be encountered at middle school and high school, with a focus on exam style example questions and preparing for standardized tests. We will also cover problem-solving questions. Each question is followed by a worked solution.

How to solve Venn diagram questions

Venn diagram questions 6th grade, venn diagram questions 7th grade, venn diagram questions 8th grade, lower ability venn diagram questions, middle ability high school venn diagram questions.

  • Looking for more Venn diagram math questions for middle and high school students?

In middle school, sets and set notation are introduced when working with Venn diagrams. A set is a collection of objects. We identify a set using braces. For example, if set A contains the odd numbers between 1 and 10, then we can write this as: 

A = {1, 3, 5, 7, 9}

Venn diagrams sort objects, called elements, into two or more sets.

Venn Diagram example

This diagram shows the set of elements 

{1,2,3,4,5,6,7,8,9,10} sorted into the following sets.

Set A= factors of 10 

Set B= even numbers

The numbers in the overlap (intersection) belong to both sets. Those that are not in set A or set B are shown outside of the circles.

Different sections of a Venn diagram are denoted in different ways.

ξ represents the whole set, called the universal set.

∅ represents the empty set, a set containing no elements.

Venn Diagrams Check for Understanding Quiz

Wondering if your students have fully grasped Venn diagrams? Use this quiz to check their understanding across 10 questions with answers covering all things Venn diagrams!

Let’s check out some other set notation examples!

In middle school and high school, we often use Venn diagrams to establish probabilities.

We do this by reading information from the Venn diagram and applying the following formula.

For Venn diagrams we can say

Middle School Venn diagram questions

In middle school, students learn to use set notation with Venn diagrams and start to find probabilities using Venn diagrams. The questions below are examples of questions that students may encounter in 6th, 7th and 8th grade.

A question on Venn diagrams from third space learning online tutoring

1. This Venn diagram shows information about the number of people who have brown hair and the number of people who wear glasses.

15 Venn Diagram Questions Blog Question 1

How many people have brown hair and glasses?

GCSE Quiz False

The intersection, where the Venn diagrams overlap, is the part of the Venn diagram which represents brown hair AND glasses. There are 4 people in the intersection.

2. Which set of objects is represented by the Venn diagram below?

15 Venn Diagram Questions Question 2 Image 1

We can see from the Venn diagram that there are two green triangles, one triangle that is not green, three green shapes that are not triangles and two shapes that are not green or triangles. These shapes belong to set D.

3. Max asks 40 people whether they own a cat or a dog. 17 people own a dog, 14 people own a cat and 7 people own a cat and a dog. Choose the correct representation of this information on a Venn diagram.

Venn Diagram Symbols GCSE Question 3 Option A

There are 7 people who own a cat and a dog. Therefore, there must be 7 more people who own a cat, to make a total of 14 who own a cat, and 10 more people who own a dog, to make a total of 17 who own a dog.

Once we put this information on the Venn diagram, we can see that there are 7+7+10=24 people who own a cat, a dog or both.

40-24=16 , so there are 16 people who own neither.

4. The following Venn diagrams each show two sets, set A and set B . On which Venn diagram has A ′ been shaded?

15 Venn Diagram Questions Question 4 Option A

\mathrm{A}^{\prime} means not in \mathrm{A} . This is shown in diagram \mathrm{B.}

5. Place these values onto the following Venn diagram and use your diagram to find the number of elements in the set \text{S} \cup \text{O}.

\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \text{S} = square numbers \text{O} = odd numbers

15 Venn Diagram Questions Question 5 Image 1

\text{S} \cup \text{O} is the union of \text{S} or \text{O} , so it includes any element in \text{S} , \text{O} or both. The total number of elements in \text{S} , \text{O} or both is 6.

6. The Venn diagram below shows a set of numbers that have been sorted into prime numbers and even numbers.

15 Venn Diagram Questions Question 6 Image 1

A number is chosen at random. Find the probability that the number is prime and not even.

The section of the Venn diagram representing prime and not even is shown below.

15 Venn Diagram Questions Question 6 Image 2

There are 3 numbers in the relevant section out of a possible 10 numbers altogether. The probability, as a fraction, is \frac{3}{10}.

7. Some people visit the theater. The Venn diagram shows the number of people who bought ice cream and drinks in the interval.

15 Venn Diagram Questions Question 7

Ice cream is sold for $3 and drinks are sold for $ 2. A total of £262 is spent. How many people bought both a drink and an ice cream?

Money spent on drinks: 32 \times \$2 = \$64

Money spent on ice cream: 16 \times \$3 = \$48

\$64+\$48=\$112 , so the information already on the Venn diagram represents \$112 worth of sales.

\$262-\$112 = \$150 , so another \$150 has been spent.

If someone bought a drink and an ice cream, they would have spent \$2+\$3 = \$5.

\$150 \div \$5=30 , so 30 people bought a drink and an ice cream.

High school Venn diagram questions

In high school, students are expected to be able to take information from word problems and put it onto a Venn diagram involving two or three sets. The use of set notation is extended and the probabilities become more complex.

In advanced math classes, Venn diagrams are used to calculate conditional probability.

8. 50 people are asked whether they have been to France or Spain.

18 people have been to France. 23 people have been to Spain. 6 people have been to both.

By representing this information on a Venn diagram, find the probability that a person chosen at random has not been to Spain or France.

15 Venn Diagram Questions Question 8 Image 1

6 people have been to both France and Spain. This means 17 more have been to Spain to make 23  altogether, and 12 more have been to France to make 18 altogether. This makes 35 who have been to France, Spain or both and therefore 15 who have been to neither.

The probability that a person chosen at random has not been to France or Spain is \frac{15}{50}.

9. Some people were asked whether they like running, cycling or swimming. The results are shown in the Venn diagram below.

15 Venn Diagram Questions Question 9 Image 1

One person is chosen at random. What is the probability that the person likes running and cycling?

15 Venn Diagram Questions Question 9 Image 2

9 people like running and cycling (we include those who also like swimming) out of 80 people altogether. The probability that a person chosen at random likes running and cycling is \frac{9}{80}.

10. ξ = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}

\text{A} = \{ even numbers \}

\text{B} = \{ multiples of 3 \}

By completing the following Venn diagram, find \text{P}(\text{A} \cup \text{B}^{\prime}).

15 Venn Diagram Questions Question 10 Image 1

\text{A} \cup \text{B}^{\prime} means \text{A} or not \text{B} . We need to include everything that is in \text{A} or is not in \text{B} . There are 13 elements in \text{A} or not in \text{B} out of a total of 16 elements.

Therefore \text{P}(\text{A} \cup \text{B}^{\prime}) = \frac{13}{16}.

11. ξ = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}

A = \{ multiples of 2 \}

By putting this information onto the following Venn diagram, list all the elements of B.

15 Venn Diagram Questions Question 11 Image 1

We can start by placing the elements in \text{A} \cap \text{B} , which is the intersection.

15 Venn Diagram Questions Question 11 Image 2

We can then add any other multiples of 2 to set \text{A}.

15 Venn Diagram Questions Question 11 Image 3

Next, we can add any unused elements from \text{A} \cup \text{B} to \text{B}.

15 Venn Diagram Questions Question 11 Image 4

Finally, any other elements can be added to the outside of the Venn diagram.

15 Venn Diagram Questions Question 11 Image 3

The elements of \text{B} are \{1, 2, 3, 4, 6, 12\}.

12. Some people were asked whether they like strawberry ice cream or chocolate ice cream. 82% said they like strawberry ice cream and 70% said they like chocolate ice cream. 4% said they like neither.

By putting this information onto a Venn diagram, find the percentage of people who like both strawberry and chocolate ice cream.

15 Venn Diagram Questions Question 12 Image 1

Here, the percentages add up to 156\%. This is 56\% too much. In this total, those who like chocolate and strawberry have been counted twice and so 56\% is equal to the number who like both chocolate and strawberry. We can place 56\% in the intersection, \text{C} \cap \text{S}

We know that the total percentage who like chocolate is 70\%, so 70-56 = 14\%-14\% like just chocolate. Similarly, 82\% like strawberry, so 82-56 = 26\%-26\% like just strawberry.

15 Venn Diagram Questions Question 12 Image 2

13. The Venn diagram below shows some information about the height and gender of 40 students.

15 Venn Diagram Questions Question 13 Image 1

A student is chosen at random. Find the probability that the student is female given that they are over 1.2 m .

We are told the student is over 1.2m. There are 20 students who are over 1.2m and 9 of them are female. Therefore the probability that the student is female given they are over 1.2m is   \frac{9}{20}.

15 Venn Diagram Questions Question 13 Image 2

14. The Venn diagram below shows information about the number of students who study history and geography.

H = history

G = geography

problem solving of venn diagram

Work out the probability that a student chosen at random studies only history.

We are told that there are 100 students in total. Therefore:

x = 12 or x = -3 (not valid) If x = 12, then the number of students who study only history is 12, and the number who study only geography is 24. The probability that a student chosen at random studies only history is \frac{12}{100}.

15. 50 people were asked whether they like camping, holiday home or hotel holidays.

18\% of people said they like all three. 7 like camping and holiday homes but not hotels. 11 like camping and hotels. \frac{13}{25} like camping.

Of the 27 who like holiday homes, all but 1 like at least one other type of holiday. 7 people do not like any of these types of holiday.

By representing this information on a Venn diagram, find the probability that a person chosen at random likes hotels given that they like holiday homes.

15 Venn Diagram Questions Question 15 Image 1

Put this information onto a Venn diagram.

15 Venn Diagram Questions Question 15 Image 2

We are told that the person likes holiday homes. There are 27 people who like holiday homes. 19 of these also like hotels. Therefore, the probability that the person likes hotels given that they like holiday homes is \frac{19}{27}.

Looking for more Venn diagram math questions for middle and high school students ?

  • Probability questions
  • Ratio questions
  • Algebra questions
  • Trigonometry questions
  • Long division questions
  • Pythagorean theorem questions

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

The content in this article was originally written by secondary teacher Beki Christian and has since been revised and adapted for US schools by elementary math teacher Katie Keeton.

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Venn Diagram Examples, Problems and Solutions

On this page:

  • What is Venn diagram? Definition and meaning.
  • Venn diagram formula with an explanation.
  • Examples of 2 and 3 sets Venn diagrams: practice problems with solutions, questions, and answers.
  • Simple 4 circles Venn diagram with word problems.
  • Compare and contrast Venn diagram example.

Let’s define it:

A Venn Diagram is an illustration that shows logical relationships between two or more sets (grouping items). Venn diagram uses circles (both overlapping and nonoverlapping) or other shapes.

Commonly, Venn diagrams show how given items are similar and different.

Despite Venn diagram with 2 or 3 circles are the most common type, there are also many diagrams with a larger number of circles (5,6,7,8,10…). Theoretically, they can have unlimited circles.

Venn Diagram General Formula

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

Don’t worry, there is no need to remember this formula, once you grasp the meaning. Let’s see the explanation with an example.

This is a very simple Venn diagram example that shows the relationship between two overlapping sets X, Y.

X – the number of items that belong to set A Y – the number of items that belong to set B Z – the number of items that belong to set A and B both

From the above Venn diagram, it is quite clear that

n(A) = x + z n(B) = y + z n(A ∩ B) = z n(A ∪ B) = x +y+ z.

Now, let’s move forward and think about Venn Diagrams with 3 circles.

Following the same logic, we can write the formula for 3 circles Venn diagram :

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)

Venn Diagram Examples (Problems with Solutions)

As we already know how the Venn diagram works, we are going to give some practical examples (problems with solutions) from the real life.

2 Circle Venn Diagram Examples (word problems):

Suppose that in a town, 800 people are selected by random types of sampling methods . 280 go to work by car only, 220 go to work by bicycle only and 140 use both ways – sometimes go with a car and sometimes with a bicycle.

Here are some important questions we will find the answers:

  • How many people go to work by car only?
  • How many people go to work by bicycle only?
  • How many people go by neither car nor bicycle?
  • How many people use at least one of both transportation types?
  • How many people use only one of car or bicycle?

The following Venn diagram represents the data above:

Now, we are going to answer our questions:

  • Number of people who go to work by car only = 280
  • Number of people who go to work by bicycle only = 220
  • Number of people who go by neither car nor bicycle = 160
  • Number of people who use at least one of both transportation types = n(only car) + n(only bicycle) + n(both car and bicycle) = 280 + 220 + 140 = 640
  • Number of people who use only one of car or bicycle = 280 + 220 = 500

Note: The number of people who go by neither car nor bicycle (160) is illustrated outside of the circles. It is a common practice the number of items that belong to none of the studied sets, to be illustrated outside of the diagram circles.

We will deep further with a more complicated triple Venn diagram example.

3 Circle Venn Diagram Examples:

For the purposes of a marketing research , a survey of 1000 women is conducted in a town. The results show that 52 % liked watching comedies, 45% liked watching fantasy movies and 60% liked watching romantic movies. In addition, 25% liked watching comedy and fantasy both, 28% liked watching romantic and fantasy both and 30% liked watching comedy and romantic movies both. 6% liked watching none of these movie genres.

Here are our questions we should find the answer:

  • How many women like watching all the three movie genres?
  • Find the number of women who like watching only one of the three genres.
  • Find the number of women who like watching at least two of the given genres.

Let’s represent the data above in a more digestible way using the Venn diagram formula elements:

  • n(C) = percentage of women who like watching comedy = 52%
  • n(F ) = percentage of women who like watching fantasy = 45%
  • n(R) = percentage of women who like watching romantic movies= 60%
  • n(C∩F) = 25%; n(F∩R) = 28%; n(C∩R) = 30%
  • Since 6% like watching none of the given genres so, n (C ∪ F ∪ R) = 94%.

Now, we are going to apply the Venn diagram formula for 3 circles. 

94% = 52% + 45% + 60% – 25% – 28% – 30% + n (C ∩ F ∩ R)

Solving this simple math equation, lead us to:

n (C ∩ F ∩ R)  = 20%

It is a great time to make our Venn diagram related to the above situation (problem):

See, the Venn diagram makes our situation much more clear!

From the Venn diagram example, we can answer our questions with ease.

  • The number of women who like watching all the three genres = 20% of 1000 = 200.
  • Number of women who like watching only one of the three genres = (17% + 12% + 22%) of 1000 = 510
  • The number of women who like watching at least two of the given genres = (number of women who like watching only two of the genres) +(number of women who like watching all the three genres) = (10 + 5 + 8 + 20)% i.e. 43% of 1000 = 430.

As we mentioned above 2 and 3 circle diagrams are much more common for problem-solving in many areas such as business, statistics, data science and etc. However, 4 circle Venn diagram also has its place.

4 Circles Venn Diagram Example:

A set of students were asked to tell which sports they played in school.

The options are: Football, Hockey, Basketball, and Netball.

Here is the list of the results:

The next step is to draw a Venn diagram to show the data sets we have.

It is very clear who plays which sports. As you see the diagram also include the student who does not play any sports (Dorothy) by putting her name outside of the 4 circles.

From the above Venn diagram examples, it is obvious that this graphical tool can help you a lot in representing a variety of data sets. Venn diagram also is among the most popular types of graphs for identifying similarities and differences .

Compare and Contrast Venn Diagram Example:

The following compare and contrast example of Venn diagram compares the features of birds and bats:

Tools for creating Venn diagrams

It is quite easy to create Venn diagrams, especially when you have the right tool. Nowadays, one of the most popular way to create them is with the help of paid or free graphing software tools such as:

You can use Microsoft products such as:

Some free mind mapping tools are also a good solution. Finally, you can simply use a sheet of paper or a whiteboard.

Conclusion:

A Venn diagram is a simple but powerful way to represent the relationships between datasets. It makes understanding math, different types of data analysis , set theory and business information easier and more fun for you.

Besides of using Venn diagram examples for problem-solving and comparing, you can use them to present passion, talent, feelings, funny moments and etc.

Be it data science or real-world situations, Venn diagrams are a great weapon in your hand to deal with almost any kind of information.

If you need more chart examples, our posts fishbone diagram examples and what does scatter plot show might be of help.

About The Author

problem solving of venn diagram

Silvia Valcheva

Silvia Valcheva is a digital marketer with over a decade of experience creating content for the tech industry. She has a strong passion for writing about emerging software and technologies such as big data, AI (Artificial Intelligence), IoT (Internet of Things), process automation, etc.

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Venn Diagrams: Exercises

Intro Set Not'n Sets Exercises Diag. Exercises

Venn diagram word problems generally give you two or three classifications and a bunch of numbers. You then have to use the given information to populate the diagram and figure out the remaining information. For instance:

Out of forty students, 14 are taking English Composition and 29 are taking Chemistry.

  • If five students are in both classes, how many students are in neither class?
  • How many are in either class?
  • What is the probability that a randomly-chosen student from this group is taking only the Chemistry class?

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There are two classifications in this universe: English students and Chemistry students.

First I'll draw my universe for the forty students, with two overlapping circles labelled with the total in each:

(Well, okay; they're ovals, but they're always called "circles".)

Five students are taking both classes, so I'll put " 5 " in the overlap:

I've now accounted for five of the 14 English students, leaving nine students taking English but not Chemistry, so I'll put " 9 " in the "English only" part of the "English" circle:

I've also accounted for five of the 29 Chemistry students, leaving 24 students taking Chemistry but not English, so I'll put " 24 " in the "Chemistry only" part of the "Chemistry" circle:

This tells me that a total of 9 + 5 + 24 = 38 students are in either English or Chemistry (or both). This gives me the answer to part (b) of this exercise. This also leaves two students unaccounted for, so they must be the ones taking neither class, which is the answer to part (a) of this exercise. I'll put " 2 " inside the box, but outside the two circles:

The last part of this exercise asks me for the probability that a agiven student is taking Chemistry but not English. Out of the forty students, 24 are taking Chemistry but not English, which gives me a probability of:

24/40 = 0.6 = 60%

  • Two students are taking neither class.
  • There are 38 students in at least one of the classes.
  • There is a 60% probability that a randomly-chosen student in this group is taking Chemistry but not English.

Years ago, I discovered that my (now departed) cat had a taste for the adorable little geckoes that lived in the bushes and vines in my yard, back when I lived in Arizona. In one month, suppose he deposited the following on my carpet:

  • six gray geckoes,
  • twelve geckoes that had dropped their tails in an effort to escape capture, and
  • fifteen geckoes that he'd chewed on a little

In addition:

  • only one of the geckoes was gray, chewed-on, and tailless;
  • two were gray and tailless but not chewed-on;
  • two were gray and chewed-on but not tailless.

If there were a total of 24 geckoes left on my carpet that month, and all of the geckoes were at least one of "gray", "tailless", and "chewed-on", how many were tailless and chewed-on, but not gray?

If I work through this step-by-step, using what I've been given, I can figure out what I need in order to answer the question. This is a problem that takes some time and a few steps to solve.

They've given me that each of the geckoes had at least one of the characteristics, so each is a member of at least one of the circles. This means that there will be nothing outside of the circles; the circles will account for everything in this particular universe.

There was one gecko that was gray, tailless, and chewed on, so I'll draw my Venn diagram with three overlapping circles, and I'll put " 1 " in the center overlap:

Two of the geckoes were gray and tailless but not chewed-on, so " 2 " goes in the rest of the overlap between "gray" and "tailless".

Two of them were gray and chewed-on but not tailless, so " 2 " goes in the rest of the overlap between "gray" and "chewed-on".

Since a total of six were gray, and since 2 + 1 + 2 = 5 of these geckoes have already been accounted for, this tells me that there was only one left that was only gray.

This leaves me needing to know how many were tailless and chewed-on but not gray, which is what the problem asks for. But, because I don't know how many were only chewed on or only tailless, I cannot yet figure out the answer value for the remaining overlap section.

I need to work with a value that I don't yet know, so I need a variable. I'll let " x " stand for this unknown number of tailless, chewed-on geckoes.

I do know the total number of chewed geckoes ( 15 ) and the total number of tailless geckoes ( 12 ). After subtracting, this gives me expressions for the remaining portions of the diagram:

only chewed on:

15 − 2 − 1 − x = 12 − x

only tailless:

12 − 2 − 1 − x = 9 − x

There were a total of 24 geckoes for the month, so adding up all the sections of the diagram's circles gives me: (everything from the "gray" circle) plus (the unknown from the remaining overlap) plus (the only-chewed-on) plus (the only-tailless), or:

(1 + 2 + 1 + 2) + ( x )

+ (12 − x ) + (9 − x )

= 27 − x = 24

Solving , I get x = 3 . So:

Three geckoes were tailless and chewed on but not gray.

(No geckoes or cats were injured during the production of the above word problem.)

For more word-problem examples to work on, complete with worked solutions, try this page provided by Joe Kahlig of Texas A&M University. There is also a software package (DOS-based) available through the Math Archives which can give you lots of practice with the set-theory aspect of Venn diagrams. The program is not hard to use, but you should definitely read the instructions before using.

URL: https://www.purplemath.com/modules/venndiag4.htm

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problem solving of venn diagram

Venn Diagram Word Problems

Related Pages Venn Diagrams Intersection Of Two Sets Intersection Of Three Sets More Lessons On Sets More GCSE/IGCSE Maths Lessons

In these lessons, we will learn how to solve word problems using Venn Diagrams that involve two sets or three sets. Examples and step-by-step solutions are included in the video lessons.

What Are Venn Diagrams?

Venn diagrams are the principal way of showing sets in a diagrammatic form. The method consists primarily of entering the elements of a set into a circle or ovals.

Before we look at word problems, see the following diagrams to recall how to use Venn Diagrams to represent Union, Intersection and Complement.

Venn Diagram

How To Solve Problems Using Venn Diagrams?

This video solves two problems using Venn Diagrams. One with two sets and one with three sets.

Problem 1: 150 college freshmen were interviewed. 85 were registered for a Math class, 70 were registered for an English class, 50 were registered for both Math and English.

a) How many signed up only for a Math Class? b) How many signed up only for an English Class? c) How many signed up for Math or English? d) How many signed up neither for Math nor English?

Problem 2: 100 students were interviewed. 28 took PE, 31 took BIO, 42 took ENG, 9 took PE and BIO, 10 took PE and ENG, 6 took BIO and ENG, 4 took all three subjects.

a) How many students took none of the three subjects? b) How many students took PE but not BIO or ENG? c) How many students took BIO and PE but not ENG?

How And When To Use Venn Diagrams To Solve Word Problems?

Problem: At a breakfast buffet, 93 people chose coffee and 47 people chose juice. 25 people chose both coffee and juice. If each person chose at least one of these beverages, how many people visited the buffet?

How To Use Venn Diagrams To Help Solve Counting Word Problems?

Problem: In a class of 30 students, 19 are studying French, 12 are studying Spanish and 7 are studying both French and Spanish. How many students are not taking any foreign languages?

Probability, Venn Diagrams And Conditional Probability

This video shows how to construct a simple Venn diagram and then calculate a simple conditional probability.

Problem: In a class, P(male)= 0.3, P(brown hair) = 0.5, P (male and brown hair) = 0.2 Find (i) P(female) (ii) P(male| brown hair) (iii) P(female| not brown hair)

Venn Diagrams With Three Categories

Example: A group of 62 students were surveyed, and it was found that each of the students surveyed liked at least one of the following three fruits: apricots, bananas, and cantaloupes.

34 liked apricots. 30 liked bananas. 33 liked cantaloupes. 11 liked apricots and bananas. 15 liked bananas and cantaloupes. 17 liked apricots and cantaloupes. 19 liked exactly two of the following fruits: apricots, bananas, and cantaloupes.

a. How many students liked apricots, but not bananas or cantaloupes? b. How many students liked cantaloupes, but not bananas or apricots? c. How many students liked all of the following three fruits: apricots, bananas, and cantaloupes? d. How many students liked apricots and cantaloupes, but not bananas?

Venn Diagram Word Problem

Here is an example on how to solve a Venn diagram word problem that involves three intersecting sets.

Problem: 90 students went to a school carnival. 3 had a hamburger, soft drink and ice-cream. 24 had hamburgers. 5 had a hamburger and a soft drink. 33 had soft drinks. 10 had a soft drink and ice-cream. 38 had ice-cream. 8 had a hamburger and ice-cream. How many had nothing? (Errata in video: 90 - (14 + 2 + 3 + 5 + 21 + 7 + 23) = 90 - 75 = 15)

Venn Diagrams With Two Categories

This video introduces 2-circle Venn diagrams, and using subtraction as a counting technique.

How To Use 3-Circle Venn Diagrams As A Counting Technique?

Learn about Venn diagrams with two subsets using regions.

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How to Solve Problems Using Venn Diagrams

Venn diagrams are visual tools often used to organize and understand sets and the relationships between them. They're named after John Venn, a British philosopher, and logician who introduced them in the 1880s. Venn diagrams are frequently used in various fields, including mathematics, statistics, logic, computer science, etc. They're handy for solving problems involving sets and subsets, intersections, unions, and complements.

How to Solve Problems Using Venn Diagrams

A Step-by-step Guide to Solving Problems Using Venn Diagrams

Here’s a step-by-step guide on how to solve problems using Venn diagrams:

Step 1: Understand the Problem

As with any problem-solving method, the first step is to understand the problem. What sets are involved? How are they related? What are you being asked to find?

Step 2: Draw the Diagram

Draw a rectangle to represent the universal set, which includes all possible elements. Each set within the universal set is represented by a circle. If there are two sets, draw two overlapping circles. If there are three sets, draw three overlapping circles, and so forth. Each section in the overlapping circles represents different intersections of the sets.

Step 3: Label the Diagram

Each circle (set) should be labeled appropriately. If you’re dealing with sets of different types of fruits, for example, one might be labeled “Apples” and another “Oranges”.

Step 4: Fill in the Values

Start filling in the values from the innermost part of the diagram (where all sets overlap) to the outer parts. This helps to avoid double-counting elements that belong to more than one set. Information provided in the problem usually tells you how many elements are in each set or section.

Step 5: Solve the Problem

Now, you can use the diagram to answer the question. This might involve counting the number of elements in a particular set or section of the diagram, or it might involve noticing patterns or relationships between the sets.

Step 6: Check Your Answer

Make sure your answer makes sense in the context of the problem and that you’ve accounted for all elements in the diagram.

by: Effortless Math Team about 8 months ago (category: Articles )

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Venn Diagram

A Venn diagram is used to visually represent the differences and the similarities between two concepts. Venn diagrams are also called logic or set diagrams and are widely used in set theory, logic, mathematics, businesses, teaching, computer science, and statistics.

Let's learn about Venn diagrams, their definition, symbols, and types with solved examples.

What is a Venn Diagram?

A Venn diagram is a diagram that helps us visualize the logical relationship between sets and their elements and helps us solve examples based on these sets. A Venn diagram typically uses intersecting and non-intersecting circles (although other closed figures like squares may be used) to denote the relationship between sets.

Venn diagram definition

Venn Diagram Example

Let us observe a Venn diagram example. Here is the Venn diagram that shows the correlation between the following set of numbers.

  • One set contains even numbers from 1 to 25 and the other set contains the numbers in the 5x table from 1 to 25.
  • The intersecting part shows that 10 and 20 are both even numbers and also multiples of 5 between 1 to 25.

Venn diagram example

Terms Related to Venn Diagram

Let us understand the following terms and concepts related to Venn Diagram, to understand it better.

Universal Set

Whenever we use a set, it is easier to first consider a larger set called a universal set that contains all of the elements in all of the sets that are being considered. Whenever we draw a Venn diagram:

  • A large rectangle is used to represent the universal set and it is usually denoted by the symbol E or sometimes U.
  • All the other sets are represented by circles or closed figures within this larger rectangle .
  • Every set is the subset of the universal set U.

Universal Set example

Consider the above-given image:

  • U is the universal set with all the numbers 1-10, enclosed within the rectangle.
  • A is the set of even numbers 1-10, which is the subset of the universal set U and it is placed inside the rectangle.
  • All the numbers between 1-10, that are not even, will be placed outside the circle and within the rectangle as shown above.

Venn diagrams are used to show subsets. A subset is actually a set that is contained within another set. Let us consider the examples of two sets A and B in the below-given figure. Here, A is a subset of B. Circle A is contained entirely within circle B. Also, all the elements of A are elements of set B.

Venn diagram to represent subsets and supersets

This relationship is symbolically represented as A ⊆ B. It is read as A is a subset of B or A subset B. Every set is a subset of itself. i.e. A ⊆ A. Here is another example of subsets :

  • N = set of natural numbers
  • I = set of integers
  • Here N ⊂ I, because all-natural numbers are integers .

Venn Diagram Symbols

There are more than 30 Venn diagram symbols. We will learn about the three most commonly used symbols in this section. They are listed below as:

Let us understand the concept and the usage of the three basic Venn diagram symbols using the image given below.

Venn diagram example

Venn Diagram for Sets Operations

In set theory, we can perform certain operations on given sets. These operations are as follows,

  • Union of Set
  • Intersection of set
  • Complement of set
  • Difference of set

Union of Sets Venn Diagram

The union of two sets A and B can be given by: A ∪ B = {x | x ∈ A or x ∈ B}. This operation on the elements of set A and B can be represented using a Venn diagram with two circles. The total region of both the circles combined denotes the union of sets A and B.

Intersection of Set Venn Diagram

The intersection of sets, A and B is given by: A ∩ B = {x : x ∈ A and x ∈ B}. This operation on set A and B can be represented using a Venn diagram with two intersecting circles. The region common to both the circles denotes the intersection of set A and Set B.

Complement of Set Venn Diagram

The complement of any set A can be given as A'. This represents elements that are not present in set A and can be represented using a Venn diagram with a circle. The region covered in the universal set, excluding the region covered by set A, gives the complement of A.

Difference of Set Venn Diagram

The difference of sets can be given as, A - B. It is also referred to as a ‘relative complement’. This operation on sets can be represented using a Venn diagram with two circles. The region covered by set A, excluding the region that is common to set B, gives the difference of sets A and B.

We can observe the above-explained operations on sets using the figures given below,

sets operations and venn diagrams

Venn Diagram for Three Sets

Three sets Venn diagram is made up of three overlapping circles and these three circles show how the elements of the three sets are related. When a Venn diagram is made of three sets, it is also called a 3-circle Venn diagram. In a Venn diagram, when all these three circles overlap, the overlapping parts contain elements that are either common to any two circles or they are common to all the three circles. Let us consider the below given example:

Venn diagram for three sets

Here are some important observations from the above image:

  • Elements in P and Q = elements in P and Q only plus elements in P, Q, and R.
  • Elements in Q and R = elements in Q and R only plus elements in P, Q, and R.
  • Elements in P and R = elements in P and R only plus elements in P, Q, and R.

How to Draw a Venn Diagram?

Venn diagrams can be drawn with unlimited circles. Since more than three becomes very complicated, we will usually consider only two or three circles in a Venn diagram. Here are the 4 easy steps to draw a Venn diagram:

  • Step 1: Categorize all the items into sets.
  • Step 2: Draw a rectangle and label it as per the correlation between the sets.
  • Step 3: Draw the circles according to the number of categories you have.
  • Step 4: Place all the items in the relevant circles.

Example: Let us draw a Venn diagram to show categories of outdoor and indoor for the following pets: Parrots, Hamsters, Cats, Rabbits, Fish, Goats, Tortoises, Horses.

  • Step 1: Categorize all the items into sets (Here, its pets): Indoor pets: Cats, Hamsters, and, Parrots. Outdoor pets: Horses, Tortoises, and Goats. Both categories (outdoor and indoor): Rabbits and Fish.
  • Step 2: Draw a rectangle and label it as per the correlation between the two sets. Here, let's label the rectangle as Pets.
  • Step 3: Draw the circles according to the number of categories you have. There are two categories in the sample question: outdoor pets and indoor pets. So, let us draw two circles and make sure the circles overlap.

Venn diagram example 1

  • Step 4: Place all the pets in the relevant circles. If there are certain pets that fit both the categories, then place them at the intersection of sets , where the circles overlap. Rabbits and fish can be kept as indoor and outdoor pets, and hence they are placed at the intersection of both circles.

Venn diagram example 2

  • Step 5: If there is a pet that doesn't fit either the indoor or outdoor sets, then place it within the rectangle but outside the circles.

Venn Diagram Formula

For any two given sets A and B, the Venn diagram formula is used to find one of the following: the number of elements of A, B, A U B, or A ⋂ B when the other 3 are given. The formula says:

n(A U B) = n(A) + n(B) – n (A ⋂ B)

Here, n(A) and n(B) represent the number of elements in A and B respectively. n(A U B) and n(A ⋂ B) represent the number of elements in A U B and A ⋂ B respectively. This formula is further extended to 3 sets as well and it says:

  • n (A U B U C) = n(A) + n(B) + n(C) - n(A ⋂ B) - n(B ⋂ C) - n(C ⋂ A) + n(A ⋂ B ⋂ C)

Here is an example of Venn diagram formula.

Example:  In a cricket school, 12 players like bowling, 15 like batting, and 5 like both. Then how many players like either bowling or batting.

Let A and B be the sets of players who like bowling and batting respectively. Then

n(A ⋂ B) = 5

We have to find n(A U B). Using the Venn diagram formula,

n(A U B) = 12 + 15 - 5 = 22.

Applications of Venn Diagram

There are several advantages to using Venn diagrams. Venn diagram is used to illustrate concepts and groups in many fields, including statistics, linguistics, logic, education, computer science, and business.

  • We can visually organize information to see the relationship between sets of items, such as commonalities and differences, and to depict the relations for visual communication.
  • We can compare two or more subjects and clearly see what they have in common versus what makes them different. This might be done for selecting an important product or service to buy.
  • Mathematicians also use Venn diagrams in math to solve complex equations.
  • We can use Venn diagrams to compare data sets and to find correlations .
  • Venn diagrams can be used to reason through the logic behind statements or equations .

☛ Related Articles:

Check out the following pages related to Venn diagrams:

  • Operations on Sets
  • Roster Notation
  • Set Builder Notation
  • Probability

Important Notes on Venn Diagrams:

Here is a list of a few points that should be remembered while studying Venn diagrams:

  • Every set is a subset of itself i.e., A ⊆ A.
  • A universal set accommodates all the sets under consideration.
  • If A ⊆ B and B ⊆ A, then A = B
  • The complement of a complement is the given set itself.

Examples of Venn Diagram

Example 1: Let us take an example of a set with various types of fruits, A = {guava, orange, mango, custard apple, papaya, watermelon, cherry}. Represent these subsets using sets notation: a) Fruit with one seed b) Fruit with more than one seed

Solution: Among the various types of fruit, only mango and cherry have one seed.

Answer:    a) Fruit with one seed = {mango, cherry}  b) Fruit with more than one seed = {guava, orange, custard apple, papaya, watermelon}

Note:  If we represent these two sets on a Venn diagram, the intersection portion is empty.

Example 2: Let us take an example of two sets A and B, where A = {3, 7, 9} and B = {4, 8}. These two sets are subsets of the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Find A ∪ B.

Solution: The Venn diagram for the above relations can be drawn as:

Venn Diagram solved examples

Answer:  A ∪ B means, all the elements that belong to either set A or set B or both the sets = {3, 4, 7, 8, 9}

Example 3: Using Venn diagram, find X ∩ Y, given that X = {1, 3, 5}, Y = {2, 4, 6}.

Given: X = {1, 3, 5}, Y = {2, 4, 6}

The Venn diagram for the above example can be given as,

venn diagram example

Answer:  From the blue shaded portion of Venn diagram, we observe that, X ∩ Y = ∅ ( null set ).

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problem solving of venn diagram

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Venn Diagram Practice Questions

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FAQs on Venn Diagrams

What is a venn diagram in math.

In math, a Venn diagram is used to visualize the logical relationship between sets and their elements and helps us solve examples based on these sets.

How do You Read a Venn Diagram?

These are steps to be followed while reading a Venn diagram:

  • First, observe all the circles that are present in the entire diagram.
  • Every element present in a circle is its own item or data set.
  • The intersecting or the overlapping portions of the circles contain the items that are common to the different circles.
  • The parts that do not overlap or intersect show the elements that are unique to the different circle.

What is the Importance of Venn Diagram?

Venn diagrams are used in different fields including business, statistics, linguistics, etc. Venn diagrams can be used to visually organize information to see the relationship between sets of items, such as commonalities and differences, and to depict the relations for visual communication.

What is the Middle of a Venn Diagram Called?

When two or more sets intersect, overlap in the middle of a Venn diagram, it is called the intersection of a Venn diagram. This intersection contains all the elements that are common to all the different sets that overlap.

How to Represent a Universal Set Using Venn Diagram?

A large rectangle is used to represent the universal set and it is usually denoted by the symbol E or sometimes U. All the other sets are represented by circles or closed figures within this larger rectangle that represents the universal set.

What are the Different Types of Venn Diagrams?

The different types of Venn diagrams are:

  • Two-set Venn diagram: The simplest of the Venn diagrams, that is made up of two circles or ovals of different sets to show their overlapping properties.
  • Three-set Venn diagram: These are also called the three-circle Venn diagram, as they are made using three circles.
  • Four-set Venn diagram: These are made out of four overlapping circles or ovals.
  • Five-set Venn diagram: These comprise of five circles, ovals, or curves. In order to make a five-set Venn diagram, you can also pair a three-set diagram with repeating curves or circles.

What are the Different Fields of Applications of Venn Diagrams?

There are different cases of applications of Venn diagrams: Set theory, logic, mathematics, businesses, teaching, computer science, and statistics.

Can a Venn Diagram Have 2 Non Intersecting Circles?

Yes, a Venn digram can have two non intersecting circles where there is no data that is common to the categories belonging to both circles.

What is the Formula of Venn Diagram?

The formula that is very helpful to find the unknown information about a Venn diagram is n(A U B) = n(A) + n(B) – n (A ⋂ B), where

  • A and B are two sets.
  • n(A U B) is the number of elements in A U B.
  • n (A ⋂ B) is the number of elements in A ⋂ B.

Can a Venn Diagram Have 3 Circles?

Yes, a Venn diagram can have 3 circles , and it's called a three-set Venn diagram to show the overlapping properties of the three circles.

What is Union in the Venn Diagram?

A union is one of the basic symbols used in the Venn diagram to show the relationship between the sets. A union of two sets C and D can be shown as C ∪ D, and read as C union D. It means, the elements belong to either set C or set D or both the sets.

What is A ∩ B Venn Diagram?

A ∩ B (which means A intersection B) in the Venn diagram represents the portion that is common to both the circles related to A and B.  A ∩ B can be a null set as well and in this case, the two circles will either be non-intersecting or can be represented with intersecting circles having no data in the intersection portion.

  • 1.3 Understanding Venn Diagrams
  • Introduction
  • 1.1 Basic Set Concepts
  • 1.2 Subsets
  • 1.4 Set Operations with Two Sets
  • 1.5 Set Operations with Three Sets
  • Key Concepts
  • Formula Review
  • Chapter Review
  • Chapter Test
  • 2.1 Statements and Quantifiers
  • 2.2 Compound Statements
  • 2.3 Constructing Truth Tables
  • 2.4 Truth Tables for the Conditional and Biconditional
  • 2.5 Equivalent Statements
  • 2.6 De Morgan’s Laws
  • 2.7 Logical Arguments
  • 3.1 Prime and Composite Numbers
  • 3.2 The Integers
  • 3.3 Order of Operations
  • 3.4 Rational Numbers
  • 3.5 Irrational Numbers
  • 3.6 Real Numbers
  • 3.7 Clock Arithmetic
  • 3.8 Exponents
  • 3.9 Scientific Notation
  • 3.10 Arithmetic Sequences
  • 3.11 Geometric Sequences
  • 4.1 Hindu-Arabic Positional System
  • 4.2 Early Numeration Systems
  • 4.3 Converting with Base Systems
  • 4.4 Addition and Subtraction in Base Systems
  • 4.5 Multiplication and Division in Base Systems
  • 5.1 Algebraic Expressions
  • 5.2 Linear Equations in One Variable with Applications
  • 5.3 Linear Inequalities in One Variable with Applications
  • 5.4 Ratios and Proportions
  • 5.5 Graphing Linear Equations and Inequalities
  • 5.6 Quadratic Equations with Two Variables with Applications
  • 5.7 Functions
  • 5.8 Graphing Functions
  • 5.9 Systems of Linear Equations in Two Variables
  • 5.10 Systems of Linear Inequalities in Two Variables
  • 5.11 Linear Programming
  • 6.1 Understanding Percent
  • 6.2 Discounts, Markups, and Sales Tax
  • 6.3 Simple Interest
  • 6.4 Compound Interest
  • 6.5 Making a Personal Budget
  • 6.6 Methods of Savings
  • 6.7 Investments
  • 6.8 The Basics of Loans
  • 6.9 Understanding Student Loans
  • 6.10 Credit Cards
  • 6.11 Buying or Leasing a Car
  • 6.12 Renting and Homeownership
  • 6.13 Income Tax
  • 7.1 The Multiplication Rule for Counting
  • 7.2 Permutations
  • 7.3 Combinations
  • 7.4 Tree Diagrams, Tables, and Outcomes
  • 7.5 Basic Concepts of Probability
  • 7.6 Probability with Permutations and Combinations
  • 7.7 What Are the Odds?
  • 7.8 The Addition Rule for Probability
  • 7.9 Conditional Probability and the Multiplication Rule
  • 7.10 The Binomial Distribution
  • 7.11 Expected Value
  • 8.1 Gathering and Organizing Data
  • 8.2 Visualizing Data
  • 8.3 Mean, Median and Mode
  • 8.4 Range and Standard Deviation
  • 8.5 Percentiles
  • 8.6 The Normal Distribution
  • 8.7 Applications of the Normal Distribution
  • 8.8 Scatter Plots, Correlation, and Regression Lines
  • 9.1 The Metric System
  • 9.2 Measuring Area
  • 9.3 Measuring Volume
  • 9.4 Measuring Weight
  • 9.5 Measuring Temperature
  • 10.1 Points, Lines, and Planes
  • 10.2 Angles
  • 10.3 Triangles
  • 10.4 Polygons, Perimeter, and Circumference
  • 10.5 Tessellations
  • 10.7 Volume and Surface Area
  • 10.8 Right Triangle Trigonometry
  • 11.1 Voting Methods
  • 11.2 Fairness in Voting Methods
  • 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
  • 11.4 Apportionment Methods
  • 11.5 Fairness in Apportionment Methods
  • 12.1 Graph Basics
  • 12.2 Graph Structures
  • 12.3 Comparing Graphs
  • 12.4 Navigating Graphs
  • 12.5 Euler Circuits
  • 12.6 Euler Trails
  • 12.7 Hamilton Cycles
  • 12.8 Hamilton Paths
  • 12.9 Traveling Salesperson Problem
  • 12.10 Trees
  • 13.1 Math and Art
  • 13.2 Math and the Environment
  • 13.3 Math and Medicine
  • 13.4 Math and Music
  • 13.5 Math and Sports
  • A | Co-Req Appendix: Integer Powers of 10

Learning Objectives

After completing this section, you should be able to:

  • Utilize a universal set with two sets to interpret a Venn diagram.
  • Utilize a universal set with two sets to create a Venn diagram.
  • Determine the complement of a set.

Have you ever ordered a new dresser or bookcase that required assembly? When your package arrives you excitedly open it and spread out the pieces. Then you check the assembly guide and verify that you have all the parts required to assemble your new dresser. Now, the work begins. Luckily for you, the assembly guide includes step-by-step instructions with images that show you how to put together your product. If you are really lucky, the manufacturer may even provide a URL or QR code connecting you to an online video that demonstrates the complete assembly process. We can likely all agree that assembly instructions are much easier to follow when they include images or videos, rather than just written directions. The same goes for the relationships between sets.

Interpreting Venn Diagrams

Venn diagrams are the graphical tools or pictures that we use to visualize and understand relationships between sets. Venn diagrams are named after the mathematician John Venn, who first popularized their use in the 1880s. When we use a Venn diagram to visualize the relationships between sets, the entire set of data under consideration is drawn as a rectangle, and subsets of this set are drawn as circles completely contained within the rectangle. The entire set of data under consideration is known as the universal set .

Consider the statement: All trees are plants. This statement expresses the relationship between the set of all plants and the set of all trees. Because every tree is a plant, the set of trees is a subset of the set of plants. To represent this relationship using a Venn diagram, the set of plants will be our universal set and the set of trees will be the subset. Recall that this relationship is expressed symbolically as: Trees ⊂ Plants . Trees ⊂ Plants . To create a Venn diagram, first we draw a rectangle and label the universal set “ U = Plants . U = Plants . ” Then we draw a circle within the universal set and label it with the word “Trees.”

This section will introduce how to interpret and construct Venn diagrams. In future sections, as we expand our knowledge of relationships between sets, we will also develop our knowledge and use of Venn diagrams to explore how multiple sets can be combined to form new sets.

Example 1.18

Interpreting the relationship between sets in a venn diagram.

Write the relationship between the sets in the following Venn diagram, in words and symbolically.

The set of terriers is a subset of the universal set of dogs. In other words, the Venn diagram depicts the relationship that all terriers are dogs. This is expressed symbolically as T ⊂ U . T ⊂ U .

Your Turn 1.18

So far, the only relationship we have been considering between two sets is the subset relationship, but sets can be related in other ways. Lions and tigers are both different types of cats, but no lions are tigers, and no tigers are lions. Because the set of all lions and the set of all tigers do not have any members in common, we call these two sets disjoint sets , or non-overlapping sets.

Two sets A A and B B are disjoint sets if they do not share any elements in common. That is, if a a is a member of set A A , then a a is not a member of set B B . If b b is a member of set B B , then b b is not a member of set A A . To represent the relationship between the set of all cats and the sets of lions and tigers using a Venn diagram, we draw the universal set of cats as a rectangle and then draw a circle for the set of lions and a separate circle for the set of tigers within the rectangle, ensuring that the two circles representing the set of lions and the set of tigers do not touch or overlap in any way.

Example 1.19

Describing the relationship between sets.

Describe the relationship between the sets in the following Venn diagram.

The set of triangles and the set of squares are two disjoint subsets of the universal set of two-dimensional figures. The set of triangles does not share any elements in common with the set of squares. No triangles are squares and no squares are triangles, but both squares and triangles are 2D figures.

Your Turn 1.19

Creating venn diagrams.

The main purpose of a Venn diagram is to help you visualize the relationship between sets. As such, it is necessary to be able to draw Venn diagrams from a written or symbolic description of the relationship between sets.

To create a Venn diagram:

  • Draw a rectangle to represent the universal set, and label it U = set name U = set name .
  • Draw a circle within the rectangle to represent a subset of the universal set and label it with the set name.

If there are multiple disjoint subsets of the universal set, their separate circles should not touch or overlap.

Example 1.20

Drawing a venn diagram to represent the relationship between two sets.

Draw a Venn diagram to represent the relationship between each of the sets.

  • All rectangles are parallelograms.
  • All women are people.
  • The set of rectangles is a subset of the set of parallelograms. First, draw a rectangle to represent the universal set and label it with U = Parallelograms U = Parallelograms , then draw a circle completely within the rectangle, and label it with the name of the set it represents, R = Rectangles R = Rectangles .

In this example, both letters and names are used to represent the sets involved, but this is not necessary. You may use either letters or names alone, as long as the relationship is clearly depicted in the diagram, as shown below.

  • The universal set is the set of people, and the set of all women is a subset of the set of people.

Your Turn 1.20

Example 1.21, drawing a venn diagram to represent the relationship between three sets.

All bicycles and all cars have wheels, but no bicycle is a car. Draw a Venn diagram to represent this relationship.

Step 1: The set of bicycles and the set of cars are both subsets of the set of things with wheels. The universal set is the set of things with wheels, so we first draw a rectangle and label it with U = Things with Wheels U = Things with Wheels .

Step 2: Because the set of bicycles and the set of cars do not share any elements in common, these two sets are disjoint and must be drawn as two circles that do not touch or overlap with the universal set.

Your Turn 1.21

The complement of a set.

Recall that if set A A is a proper subset of set U U , the universal set (written symbolically as A ⊂ U A ⊂ U ), then there is at least one element in set U U that is not in set A A . The set of all the elements in the universal set U U that are not in the subset A A is called the complement of set A A , A ' A ' . In set builder notation this is written symbolically as: A ' = { x ∈ U | x ∉ A } . A ' = { x ∈ U | x ∉ A } . The symbol ∈ ∈ is used to represent the phrase, “is a member of,” and the symbol ∉ ∉ is used to represent the phrase, “is not a member of.” In the Venn diagram below, the complement of set A A is the region that lies outside the circle and inside the rectangle. The universal set U U includes all of the elements in set A A and all of the elements in the complement of set A A , and nothing else.

Consider the set of digit numbers. Let this be our universal set, U = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . U = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . Now, let set A A be the subset of U U consisting of all the prime numbers in set U U , A = { 2 , 3 , 5 , 7 } . A = { 2 , 3 , 5 , 7 } . The complement of set A A is A ' = { 0 , 1 , 4 , 6 , 8 , 9 } . A ' = { 0 , 1 , 4 , 6 , 8 , 9 } . The following Venn diagram represents this relationship graphically.

Example 1.22

Finding the complement of a set.

For both of the questions below, A A is a proper subset of U U .

  • Given the universal set U = { Billie Eilish, Donald Glover, Bruno Mars, Adele, Ed Sheeran} U = { Billie Eilish, Donald Glover, Bruno Mars, Adele, Ed Sheeran} and set A = { Donald Glover, Bruno Mars, Ed Sheeran} A = { Donald Glover, Bruno Mars, Ed Sheeran} , find A ' . A ' .
  • Given the universal set U = { d|d is a dog } U = { d|d is a dog } and B = { b ∈ U|b is a beagle } B = { b ∈ U|b is a beagle } , find B ' . B ' .
  • The complement of set A A is the set of all elements in the universal set U U that are not in set A . A . A ' = { Billie Eilish, Adele } A ' = { Billie Eilish, Adele } .
  • The complement of set B B is the set of all dogs that are not beagles. All members of set B ′ B ′ are in the universal set because they are dogs, but they are not in set B , B , because they are not beagles. This relationship can be expressed in set build notation as follows: B ′ = { All dogs that are not beagles .} B ′ = { All dogs that are not beagles .} , B ′ = { d ∈ U | d is not a beagle .} B ′ = { d ∈ U | d is not a beagle .} , or B ′ = { d ∈ U | d ∉ B } . B ′ = { d ∈ U | d ∉ B } .

Your Turn 1.22

Check your understanding, section 1.3 exercises.

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Sets and Venn Diagrams

A set is a collection of things.

For example, the items you wear is a set: these include hat, shirt, jacket, pants, and so on.

You write sets inside curly brackets like this:

{hat, shirt, jacket, pants, ...}

You can also have sets of numbers:

  • Set of whole numbers : {0, 1, 2, 3, ...}
  • Set of prime numbers : {2, 3, 5, 7, 11, 13, 17, ...}

Ten Best Friends

You could have a set made up of your ten best friends:

  • {alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade}

Each friend is an "element" (or "member") of the set. It is normal to use lowercase letters for them.

soccer teams

Now let's say that alex, casey, drew and hunter play Soccer :

Soccer = {alex, casey, drew, hunter}

(It says the Set "Soccer" is made up of the elements alex, casey, drew and hunter.)

tennis

And casey, drew and jade play Tennis :

Tennis = {casey, drew, jade}

We can put their names in two separate circles:

You can now list your friends that play Soccer OR Tennis .

This is called a "Union" of sets and has the special symbol ∪ :

Soccer ∪ Tennis = {alex, casey, drew, hunter, jade}

Not everyone is in that set ... only your friends that play Soccer or Tennis (or both).

In other words we combine the elements of the two sets.

We can show that in a "Venn Diagram":

A Venn Diagram is clever because it shows lots of information:

  • Do you see that alex, casey, drew and hunter are in the "Soccer" set?
  • And that casey, drew and jade are in the "Tennis" set?
  • And here is the clever thing: casey and drew are in BOTH sets!

All that in one small diagram.

Intersection

"Intersection" is when you must be in BOTH sets.

In our case that means they play both Soccer AND Tennis ... which is casey and drew.

The special symbol for Intersection is an upside down "U" like this: ∩

And this is how we write it:

Soccer ∩ Tennis = {casey, drew}

In a Venn Diagram:

Which Way Does That "U" Go?

union symbol looks like cup

Think of them as "cups": ∪ holds more water than ∩ , right?

So Union ∪ is the one with more elements than Intersection ∩

You can also "subtract" one set from another.

For example, taking Soccer and subtracting Tennis means people that play Soccer but NOT Tennis ... which is alex and hunter.

Soccer − Tennis = {alex, hunter}

Summary So Far

  • ∪ is Union: is in either set or both sets
  • ∩ is Intersection: only in both sets
  • − is Difference: in one set but not the other

You can also use Venn Diagrams for 3 sets.

Let us say the third set is "Volleyball", which drew, glen and jade play:

Volleyball = {drew, glen, jade}

But let's be more "mathematical" and use a Capital Letter for each set:

  • S means the set of Soccer players
  • T means the set of Tennis players
  • V means the set of Volleyball players

The Venn Diagram is now like this:

Union of 3 Sets: S ∪ T ∪ V

You can see (for example) that:

  • drew plays Soccer, Tennis and Volleyball
  • jade plays Tennis and Volleyball
  • alex and hunter play Soccer, but don't play Tennis or Volleyball
  • no-one plays only Tennis

We can now have some fun with Unions and Intersections ...

S = {alex, casey, drew, hunter}

T ∪ V = {casey, drew, jade, glen}

S ∩ V = {drew}

And how about this ...

  • take the previous set S ∩ V
  • then subtract T :

(S ∩ V) − T = {}

Hey, there is nothing there!

That is OK, it is just the "Empty Set". It is still a set, so we use the curly brackets with nothing inside: {}

The Empty Set has no elements: {}

Universal Set

The Universal Set is the set that has everything. Well, not exactly everything. Everything that we are interested in now.

Sadly, the symbol is the letter "U" ... which is easy to confuse with the ∪ for Union. You just have to be careful, OK?

In our case the Universal Set is our Ten Best Friends.

U = {alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade}

We can show the Universal Set in a Venn Diagram by putting a box around the whole thing:

Now you can see ALL your ten best friends, neatly sorted into what sport they play (or not!).

And then we can do interesting things like take the whole set and subtract the ones who play Soccer :

We write it this way:

U − S = {blair, erin, francis, glen, ira, jade}

Which says "The Universal Set minus the Soccer Set is the Set {blair, erin, francis, glen, ira, jade}"

In other words "everyone who does not play Soccer".

And there is a special way of saying "everything that is not ", and it is called "complement" .

We show it by writing a little "C" like this:

Which means "everything that is NOT in S", like this:

S c = {blair, erin, francis, glen, ira, jade} (exactly the same as the U − S example from above)

  • A c is the Complement of A: everything that is not in A
  • Empty Set: the set with no elements. Shown by {}
  • Universal Set: all things we are interested in

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Mathematics LibreTexts

2.2: Venn Diagrams

  • Last updated
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  • Page ID 82983

  • Julie Harland
  • MiraCosta College

Screen Shot 2021-03-30 at 9.32.41 PM.png

Study the Venn diagrams on this and the following pages. It takes a whole lot of practice to shade or identify regions of Venn diagrams. Be advised that it may be necessary to shade several practice diagrams along the way before you get to the final result.

We shade Venn diagrams to represent sets. We will be doing some very easy, basic Venn diagrams as well as several involved and complicated Venn diagrams.

To find the intersection of two sets, you might try shading one region in a given direction, and another region in a different direction. Then you would look where those shadings overlap. That overlap would be the intersection.

Screen Shot 2021-03-30 at 9.41.32 PM.png

For example, to visualize \(A \cap B\), shade A with horizontal lines and B with vertical lines. Then the overlap is \(A \cap B\). The diagram on the left would be a first step in getting the answer. The shaded part on the diagram to the right shows the final answer.

Screen Shot 2021-03-30 at 9.41.37 PM.png

Here are two problems for you to try. Only shade in the final answer for each exercise.

Shade the region that represents \(A \cap C\)

Screen Shot 2021-03-30 at 9.41.24 PM.png

Shade the region that represents \(B \cap C\)

Screen Shot 2021-03-30 at 9.41.18 PM.png

To shade the union of two sets, shade each region completely or shade both regions in the same direction. Thus, to find the union of A and B , shade all of A and all of B .

The final answer is represented by the shaded area in the diagram to the right.

Shade the region that represents \(A \cup C\)

Shade the region that represents \(B \cup C\)

Screen Shot 2021-03-30 at 10.02.25 PM.png

For the complement of a region, shade everything outside the given region. You can think of it as shading everything except that region. On the Venn diagram to the left, the shaded area represents A . On the Venn diagram to the right, the shaded area represents .

Screen Shot 2021-03-30 at 10.02.30 PM.png

Many people are confused about what part of the Venn diagram represents the universe, U . The universe is the entire Venn diagram, including the sets A , B and C . The three Venn diagrams on the next page illustrate the differences between U , \(U^{c}\) and \((A \cup B \cup C)^{c}\). Carefully note these differences.

Usually, parentheses are necessary to indicate which operation needs to be done first. If there is only union or intersection involved, this isn’t necessary as in ( A \(\cup\) B \(\cup\) C )\(^{c}\) above. Convince yourself that (( A \(\cup\) B ) \(\cup\) C ) = ( A \(\cup\) ( B \(\cup\) C )). Similarly, convince yourself of the analogous fact for intersection by performing the following steps. On the first Venn diagram below, shade A \(\cap\) B with horizontal lines and shade C with vertical lines. Then, the overlap is (( A \(\cap\) B ) \(\cap\) C ). On the second Venn diagram, shade A with lines slanting to the right and B \(\cup\) C with lines slanting to the left. Then the overlap is ( A \(\cap\) ( B \(\cap\) C )). Check to see that the final answer, the overlap in this case, is the same for both. Shade the final answer in the third Venn diagram.

a. ( A \(\cap\) B ) \(\cap\) C

b. ( A \(\cup\) ( B \(\cup\) C ))

c. Shade final answer here.

Now, it's time for you to try a few more diagrams on your own. It may take more than one step to figure out the answer. You might need to do preliminary drawings on scratch paper first. The shadings you show here should be the final answer only, but you should be able to explain and support how you arrived at your answer. Compare your answers with other people in your class and make sure a consensus is reached on the correct answer. Do this for all the Venn diagrams throughout this exercise set. Shade in the region that represents what is written above each of the six Venn diagrams on the following page . Note that in cases involving more than one operation, it is necessary to use parentheses and follow order of operations. Exercises 10 and 11 illustrate why this is necessary.

( C \(\cap\) A )\(^{c}\)

( B \(\cup\) C )\(^{c}\)

( A \(\cap\) B \(\cap\) C )\(^{c}\)

Exercise 10

( A \(\cap\) B ) \(\cap\) C

Exercise 11

( A \(\cap\) ( B \(\cap\) C )

Screen Shot 2021-03-31 at 10.55.06 AM.png

For difference, shade the region coming before the difference sign ( – ) but don’t include or shade any part of the region that follows the difference sign. The Venn on the left represents A–B and the one on the right represents C – A.

Screen Shot 2021-03-31 at 11.08.29 AM.png

Exercise 12

Shade the region that represents A – C

Exercise 13

Shade the region that represents B – C

Study the following Venn diagrams. Make sure you understand how to get the answers.

Screen Shot 2021-03-31 at 11.19.46 AM.png

It's your turn to shade in the region that represents what is written above each diagram.

Exercise 14

( A \(\cap\) C ) – B

Screen Shot 2021-03-31 at 11.37.42 AM.png

Exercise 15

B – ( A \(\cap\) C )

Exercise 16

( A – C ) \(\cup\) ( B – A )

Suppose you wanted to find ((C – A ) \(\cap\) B )\(^{c}\). This would probably take a few steps to get the answer. One approach to finding the correct shading is to notice that the final answer is the complement of ( C – A) \(\cap\) B . That means we would have to first figure out what (C – A) \(\cap\) B looked like. In order to do that, we notice that this is the intersection of two things C – A and B. On the blank Venn diagram to the left below, shade C – A with horizontal lines and B with vertical lines. The overlap would be the intersection. The overlap on your drawing should match the shading shown on the Venn diagram in the middle. Does it? The last step would then be to take the complement of the shading shown on the middle diagram. This is shown on the Venn diagram on the far right. So, it took drawing three Venns to come up with the final answer for this problem. Someone else might be able to do it in fewer steps while someone else might take more steps.

Exercise 17

As mentioned previously, it takes a lot of practice to get good at shading Venn diagrams. It’s even trickier to look at a Venn diagram and describe it, In fact, there is usually more than one way to describe a Venn diagram. For example, the shading for (( C – A ) \(\cap\) B )\(^{c}\) shown on the previous page is the same as it is for (( C \(\cap\) B ) – A )\(^{c}\). What does this mean? We’re so used to only having one correct answer. Well, consider if someone asked you to write an arithmetic problem for which the answer was 2. There would be infinitely many possibilities. For example, 5 - 3 or 1 + 1 or 10/5 would all be acceptable answers. Granted, this kind of question on a test would be harder for a teacher to grade because each student’s response would have to be checked to see if it would work. There isn’t one pat answer. The same goes if a teacher asks you to look at a shading of a Venn diagram and describe it. On the other hand, if a description is given and you are asked to shade the Venn diagram, there is only one correct shading. It is much like being asked to compute an arithmetic problem. The answer to 10 - 8 is 2 and that is the only acceptable answer!

The point of all this is that to master shadings of Venn diagrams and descriptions of Venn diagrams by looking at the shadings takes lots and lots and lots of practice. Give yourself plenty of time to study and work on them and you will accomplish this feat!!!

On the next few pages, you are asked to shade several one, two and three set Venn diagrams. The correct shadings follow. Make sure you try these problems in earnest. Make sure you can explain the steps involved to arrive at the correct shading. After mastering the shadings, see if you can look at a shaded Venn diagram and come up with an accurate description. Again, remember there is more than one way to describe a given Venn diagram.

These Venn diagrams will be helpful when studying for a test. Go back and practice drawing the same Venn diagrams later. Use the answers to see if you can describe them by looking at the picture. Of course, remember that your description might not match exactly since there as more than one way to describe any given Venn diagram. If your description is different, make sure you go through the steps of shading a Venn with your description and see if your shading really matches the Venn diagram you were trying to describe.

Here are a few shaded Venn diagrams. See if you can look at the shadings and come up with a description. I’ve put some possible answers at the bottom of this page.

Here are some possible descriptions for the above Venn diagrams:

Shade the region that represents what is written above each of the one and two set Venn diagrams below. You may need to draw preliminary drawings first for some of them.

Exercise 18

Screen Shot 2021-03-31 at 11.11.42 PM.png

Exercise 19

Exercise 20, exercise 21, exercise 22.

A \(\cap\) B

Screen Shot 2021-03-31 at 11.21.25 PM.png

Exercise 23

A \(\cup\) B

Exercise 24

\(A \cup B^{c}\)

Exercise 25

\((A \cap B)^{c}\)

Exercise 26

\((A \cup B)^{c}\)

Exercise 27

( A \ B ) \(\cup\) ( B \ A )

Exercise 28

\(A^{c} \cup B^{c}\)

Exercise 29

\(A^{c} \cap B^{c}\)

Exercise 30

\((A \cup B)^{c} \cup (A \cap B)\)

Exercise 31

Exercise 32, exercise 33, exercise 34.

( A \(\cap\) B ) – C

Exercise 35

( C \(\cup\) B ) – A

Exercise 36

( A \(\cap\) B ) \(\cup\) C

Exercise 37

( A \(\cup\) B ) \(\cap\) C

Exercise 38

A \(^{c}\) – B

Exercise 39

A \(\cap\) B \(\cap\) C ) – B

Exercise 40

B – ( A \(\cup\) C )

Exercise 41

C – ( A \(\cap\) B )

Exercise 42

( B – A ) \(\cap\) ( B – C )

Exercise 43

( B – A ) \(\cup\) ( B – C )

Exercise 44

( A \(\cup\) B )\(^{c}\)

Exercise 45

A \(^{c}\) \(\cap\) B \(^{c}\)

Exercise 46

A \(^{c}\) – B \(^{c}\)

Exercise 47

( C – B) \(^{c}\)

Exercise 48

( B \(^{c}\) \(\cap\) C ) – A

Exercise 49

( A – ( B \(\cup\) C )) \(\cup\) ( B – ( A \(\cup\) C )) \(\cup\) ( C – ( A \(\cup\) B ))

Exercise 50

( A \(\cap\) C )\(^{c}\)

Exercise 51

(( A \(\cap\) B ) – C ) \(\cap\) ( C – A)

Exercise 52

A \(^{c}\) \(\cup\) C \(^{c}\)

Exercise 53

B \(\cap\) ( C \(\cup\) A \(^{c}\))

Here are the correct shadings to the exercises on the previous pages. After mastering these shadings, reverse the process by looking at the shadings on this page and try to describe them. It takes practice and patience and remember that there may be more than one way to describe some of these. In fact, many times you'll see there is a simpler way to describe them than was on the original exercise!!

Screen Shot 2021-04-01 at 7.53.19 PM.png

In the Material Card section there are blank Venn diagram templates you can use for practice.

Venn Diagram Examples for Problem Solving

Updated on: 13 September 2022

What is a Venn Diagram?

Venn diagrams define all the possible relationships between collections of sets. The most basic Venn diagrams simply consist of multiple circular boundaries describing the range of sets.

The most basic Venn diagrams - Venn Diagram Examples for Problem Solving

The overlapping areas between the two boundaries describe the elements which are common between the two, while the areas that aren’t overlapping house the elements that are different. Venn diagrams are used often in math that people tend to assume they are used only to solve math problems. But as the 3 circle Venn diagram below shows it can be used to solve many other problems.

3 circle Venn diagram is a good example of solving problems with Venn diagrams - Venn Diagram Examples for Problem Solving

Though the above diagram may look complicated, it is actually very easy to understand. Although Venn diagrams can look complex when solving business processes understanding of the meaning of the boundaries and what they stand for can simplify the process to a great extent. Let us have a look at a few examples which demonstrate how Venn diagrams can make problem solving much easier.

Example 1: Company’s Hiring Process

The first Venn diagram example demonstrates a company’s employee shortlisting process. The Human Resources department looks for several factors when short-listing candidates for a position, such as experience, professional skills and leadership competence. Now, all of these qualities are different from each other, and may or may not be present in some candidates. However, the best candidates would be those that would have all of these qualities combined.

Using Venn diagrams to find the right candidate - Venn Diagram Examples for Problem Solving

The candidate who has all three qualities is the perfect match for your organization. So by using simple Venn Diagrams like the one above, a company can easily demonstrate its hiring processes and make the selection process much easier.

A colorful and precise Venn diagram like the above can be easily created using our Venn diagram software and we have professionally designed Venn diagram templates for you to get started fast too.

Example 2: Investing in a Location

The second Venn diagram example takes things a step further and takes a look at how a company can use a Venn diagram to decide a suitable office location. The decision will be based on economic, social and environmental factors.

Venn diagram to select office location - Venn Diagram Examples for Problem Solving

In a perfect scenario you’ll find a location that has all the above factors in equal measure. But if you fail to find such a location then you can decide which factor is most important to you. Whatever the priority because you already have listed down the locations making the decision becomes easier.

Example 3: Choosing a Dream Job

The last example will reflect on how one of the life’s most complicated questions can be easily answered using a Venn diagram. Choosing a dream job is something that has stumped most college graduates, but with a single Venn diagram, this thought process can be simplified to a great extent.

First, single out the factors which matter in choosing a dream job, such as things that you love to do, things you’re good at, and finally, earning potential. Though most of us dream of being a celebrity and coming on TV, not everyone is gifted with acting skills, and that career path may not be the most viable. Instead, choosing something that you are good at, that you love to do along with something that has a good earning potential would be the most practical choice.

Venn diagram to find the dream job - Venn Diagram Examples for Problem Solving

A job which includes all of these three criteria would, therefore, be the dream job for someone. The three criteria need not necessarily be the same, and can be changed according to the individual’s requirements.

So you see, even the most complicated processes can be simplified by using these simple Venn diagrams.

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problem solving of venn diagram

Great article, and all true, but.. I hate venn diagrams! I don’t know why, they’ve just never seemed to work for me. Frustrating!

Hey thanks for writing. It helped me in many ways Thanks again 🙂

Hi Nishadha,

Nice article! I love Venn Diagrams because nothing comes to close to expressing the logical relationships between different sets of elements that well. With Microsoft Word 2003 you can create fantastic looking and colorful Venn Diagrams on the fly, with as many elements and colors as you need.

Hi Worli, Yes, Venn diagrams are a good way to solve problems, it’s a shame that it’s sort of restricted to the mathematics subject. MS Word do provides some nice options to create Venn diagrams, although it’s not the cheapest thing around.

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Venn Diagrams

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Venn Diagrams with our comprehensive guide, tailored for educators and students striving for clarity in mathematics and logic. This guide demystifies the concept of Venn Diagrams, showcasing their utility in representing logical relationships visually. Through practical examples and simplified explanations, we aim to enhance analytical thinking and problem-solving skills. Whether it’s for classroom instruction or self-study, this resource is your go-to for mastering the intricacies of Venn Diagrams.

What are Venn Diagrams?

Venn Diagrams are powerful tools used to illustrate the mathematical or logical relationship between different sets. By drawing circles that overlap, they visually represent how items or groups share common traits or differ, making complex relationships easier to understand. This method is invaluable in various fields, including statistics, logic, and education, helping students and teachers alike to dissect and comprehend the interconnectedness of concepts.

What is the Best Example of a Venn Diagram?

best example of a venn diagram

A prime example of a Venn Diagram is comparing characteristics of mammals and reptiles. By placing unique traits in separate circles and shared traits in the overlapping area, students can visually discern the similarities and differences between the two classes. This not only aids in grasping biological classifications but also in developing critical thinking skills by analyzing how groups relate to each other.

Venn Diagram Formula

Delve into the Venn Diagram formula with this succinct guide, designed to make mathematical concepts accessible for educators and students. The formula, helps quantify the elements in union and intersection of two sets, A and B. This formula is a cornerstone for understanding set theory, providing a mathematical framework to solve problems involving elements from multiple sets. Through clear examples and applications, this description empowers users to apply the Venn Diagram formula effectively in various scenarios.

n(A U B) = n(A) + n(B) – n (A ? B)

Venn diagram example.

fruits and citrus fruits

Unlock the potential of Venn Diagrams with this insightful guide, crafted to aid educators and students in visualizing complex relationships between sets. Venn Diagrams serve as a versatile tool in mathematics, logic, and beyond, illustrating intersections, unions, and differences with clarity. By providing a visual representation of how sets overlap, these diagrams facilitate a deeper understanding of concepts, enhancing analytical and critical thinking skills.

Example 1: Fruits and Citrus Fruits

A Venn Diagram showcasing the relationship between fruits in general and citrus fruits specifically. The circle for fruits includes apples, bananas, while the citrus circle includes oranges, lemons, with grapefruits appearing in the overlap, indicating it’s both a fruit and a citrus.

Example 2: Students in Sports Clubs

Illustrates students participating in basketball, football, and those who play both. Basketball and football circles overlap on students who play both, demonstrating the diagram’s ability to depict shared membership in two sets.

Example 3: Vegetarians and Vegans

This Venn Diagram differentiates between vegetarians and vegans. Vegetarians are represented in one circle, including eggs and cheese, while vegans are in another, focusing on plant-based foods. The overlap highlights foods both groups consume, like vegetables and fruits.

Example 4: Fiction and Science Fiction Books in a Library

Depicts the categorization of fiction and science fiction books. The fiction circle includes romance and historical novels, science fiction includes space operas, with dystopian novels in the overlap, indicating their dual categorization.

Example 5: Water-Soluble and Fat-Soluble Vitamins

A Venn Diagram to classify water-soluble (C and B vitamins) and fat-soluble vitamins (A, D, E, K). The diagram clearly separates them, with no overlap, indicating distinct absorption pathways in the body.

These examples underscore the Venn Diagram’s utility in teaching and learning, offering a visual method to compare and contrast various concepts across disciplines.

Terms Related to Venn Diagram

terms related to venn diagram

Dive into the world of Venn Diagrams with this essential terminology guide. Perfect for educators and students, it covers key terms necessary for understanding and applying Venn Diagrams in logical, mathematical, and statistical contexts. From “sets” to “intersections,” this guide enriches your vocabulary, enabling clearer communication and comprehension of complex relationships between groups.

  • Explanation: In a Venn Diagram, a set is represented by a circle, encompassing its elements to visualize groupings in a problem or scenario.
  • Explanation: Indicates when all elements of one set (subset) are also elements of another, shown by one circle entirely within another in a Venn Diagram.
  • Explanation: The overlapping area of circles in a Venn Diagram, representing elements shared by sets A and B.
  • Explanation: In Venn Diagrams, the union is depicted by the total area covered by both sets’ circles, including the intersection.
  • Explanation: Represented in Venn Diagrams by the area outside a set’s circle but within the universal set boundary, indicating elements excluded from the set.

Venn Diagram Symbols

Master Venn Diagram symbols with this concise guide, tailored for educational use. These symbols form the visual language of Venn Diagrams, facilitating the representation of mathematical and logical relationships between sets. Understanding these symbols is crucial for teachers and students to effectively analyze and convey information through Venn Diagrams.

  • Explanation: Each circle in a Venn Diagram encapsulates the elements of a set, serving as a visual boundary for group identification.
  • Explanation: The area where circles intersect illustrates the common elements between sets, crucial for understanding their relationships.
  • Explanation: Surrounding the circles, the rectangle defines the universe of discourse, including all elements considered in the scenario.
  • Explanation: Dots placed inside circles signify individual members of sets, making it easy to count and identify specific elements.
  • Explanation: Shaded areas outside a set but within the universal set boundary show the complement, or elements not included in the set under consideration.

Venn Diagram for Sets Operations

Explore set operations through Venn Diagrams with this insightful guide. Ideal for educational purposes, it simplifies complex set theories, making them accessible for teachers and students. Venn Diagrams visually represent set operations like unions, intersections, and complements, providing a clear method to understand and teach mathematical relationships between sets.

  • Explanation: The Venn Diagram for a union showcases the total area covered by sets A and B, emphasizing inclusivity of elements.
  • Explanation: Highlighted by the overlap between sets, the intersection focuses on elements shared by A and B.
  • Explanation: Illustrated by shading outside A but within the universal set, this operation reveals elements excluded from set A.
  • Explanation: This operation is visualized by shading parts of A’s circle that do not overlap with B, showing elements unique to A.
  • Explanation: The Venn Diagram highlights the non-overlapping parts of A and B, focusing on elements exclusive to each set, excluding their intersection.

Difference of Set Venn Diagram

Discover the essence of set differences with Venn Diagrams in this concise guide. Ideal for teachers and students, it illustrates how to visually represent the difference between two sets, A and B, by highlighting the elements that belong exclusively to set A. This concept is pivotal in understanding complex relationships in mathematics, fostering a deeper comprehension of how sets interact with each other.

  • The difference shows prime numbers that are not even, like 3 and 5, visually excluding even primes like 2.
  • Illustrates fruits that are not citrus, such as apples and bananas, by highlighting them outside the citrus section.
  • Demonstrates that non-novel books (e.g., dictionaries) belong to set A but not to set B.
  • Showcases animals that are not mammals, like birds and fish, separated visually.
  • Highlights vehicles that are not electric, indicating the broader category excludes the subset of electric vehicles.

Venn Diagram of Three Sets

venn diagram of three sets

Explore the dynamics of three-set Venn Diagrams with our guide, perfect for educators looking to depict the complex relationships between three distinct sets. This visualization tool sheds light on how sets intersect, combine, and differ, offering a comprehensive understanding of shared and unique elements. It’s an invaluable resource for enhancing analytical thinking and problem-solving skills in mathematics and logic.

  • Reveals animals that are mammals, aquatic, both, or neither, providing insight into biological classifications.
  • Depicts students who are exclusively in one subject or intersect in two or all, highlighting academic preferences.
  • Shows how books can be categorized into fiction, mystery, both, and those that are bestsellers, revealing market trends.
  • Illustrates the overlap between vegetables, foods that are green, and those high in fiber, offering dietary insights.
  • Clarifies countries that are in Europe, part of the EU, the Eurozone, or a combination, enhancing geographical understanding.

How to Draw a Venn Diagram?

Master the art of drawing Venn Diagrams with our straightforward guide. Tailored for both teachers and students, this resource simplifies the process into manageable steps, from conceptualizing set relationships to visualizing intersections and differences. Whether for mathematical equations, logical reasoning, or categorizing information, learning to draw Venn Diagrams is an essential skill for effective problem-solving and communication.

  • Begin by defining the sets you want to compare or contrast to understand their relationships.
  • Draw circles to represent each set, ensuring they overlap for common elements.
  • Clearly label each circle with the corresponding set name for easy identification.
  • Place elements in the respective areas: unique in non-overlapping and common in overlapping sections.
  • Use the completed diagram to analyze and explain the relationships between the sets, such as intersections and differences.

These guides and examples are crafted to enhance the understanding and application of Venn Diagrams in educational settings, fostering a deeper comprehension of set theory and logical analysis among students.

Applications of Venn Diagram

Venn Diagrams serve as versatile tools in education, data analysis, and problem-solving. By visually mapping out relationships between sets, they facilitate understanding of complex concepts through overlap and distinction. Ideal for both classroom learning and professional data presentation, Venn Diagrams enhance comprehension in subjects ranging from mathematics to social sciences, making them indispensable in analytical reasoning and decision-making processes.

  • Explanation: Students can visualize numbers that are exclusively prime or composite and those that share attributes with a Venn Diagram, enhancing number theory comprehension.
  • Explanation: By using Venn Diagrams, teachers can help students identify common themes and unique elements in different works, promoting deeper literary analysis.
  • Explanation: Venn Diagrams simplify the comparison of taxonomic groups, helping students grasp similarities and differences in traits among species.
  • Explanation: Businesses utilize Venn Diagrams to understand shared characteristics among different customer segments, optimizing marketing strategies.
  • Explanation: In conflict resolution, Venn Diagrams can highlight common ground and differing points, aiding in finding mutually acceptable solutions.

Venn Diagram Purpose and Benefits

Venn Diagrams are pivotal in simplifying the visualization of complex relationships, offering clarity in educational, analytical, and strategic contexts. They enable users to compare and contrast sets, highlighting similarities, differences, and intersections with ease. This visualization aids in fostering critical thinking, enhancing memory retention, and supporting effective communication, making Venn Diagrams a powerful tool for learners and professionals alike.

  • Explanation: The visual representation in Venn Diagrams aids in memory retention by simplifying abstract concepts into tangible comparisons.
  • Explanation: Venn Diagrams challenge students to think critically about how sets relate, enhancing logical reasoning skills.
  • Explanation: In group settings, Venn Diagrams serve as focal points for discussion, promoting collaborative learning and idea exchange.
  • Explanation: Venn Diagrams make it easier to present and interpret complex data, improving audience understanding in presentations.
  • Explanation: By visually comparing options, Venn Diagrams help in weighing the pros and cons, facilitating more informed decision-making.

Venn Diagram Use Cases

Venn Diagrams are employed across various fields to visually organize information, facilitating understanding and analysis of relationships between sets. From educational settings to business strategy and scientific research, they provide a clear and intuitive method to display intersections, differences, and similarities, enhancing decision-making, learning, and data interpretation.

  • Explanation: Teachers use Venn Diagrams to compare and contrast grammatical elements, helping students understand language rules.
  • Explanation: Companies employ Venn Diagrams to compare their products with competitors’, identifying unique selling points and areas for improvement.
  • Explanation: Researchers use Venn Diagrams to visualize similarities and differences in symptoms among diseases, aiding in differential diagnosis.
  • Explanation: Venn Diagrams help students understand the shared and unique characteristics of different ecosystems, promoting environmental awareness.
  • Explanation: Developers use Venn Diagrams to compare features across different software versions or competitors, guiding development priorities.

Intersection of Two Sets in Venn Diagram

Explore the concept of the intersection of two sets in Venn Diagrams, a crucial element for teachers and students in understanding shared characteristics between groups. This visualization technique marks the common elements of sets within the overlapping regions of circles, simplifying complex relationships and enhancing analytical reasoning. Perfect for classroom discussions, it facilitates a deeper comprehension of how sets interact in mathematics and logic.

  • The intersection shows students involved in both sports, highlighting shared participants in the overlapping area.
  • This intersection identifies individuals who enjoy both beverages, represented by the shared space between two circles.
  • The overlapping section illustrates animals sharing traits of mammals and carnivores, aiding in biological classification.
  • In the intersection, books categorized as fiction and mystery are shown, demonstrating how genres can overlap.
  • This example uses the intersection to show countries that are part of Europe and the Schengen Agreement, facilitating a geographic and political understanding.

Union of Two Sets in Venn Diagram

The union of two sets in Venn Diagrams represents the combination of all elements from both sets, including the shared and unique elements. This concept is vital for students and teachers, offering a visual method to comprehend the totality of distinct and overlapping characteristics within groups. By enhancing visualization skills, it aids in grasping the breadth of set relationships, making it a fundamental tool in mathematical education.

  • The union includes all books from both genres, emphasizing the extensive range of literature available.
  • This union displays students taking either or both subjects, showcasing the diverse interests within the school population.
  • The combined set includes all countries from both continents, highlighting the vast geographical coverage.
  • By uniting the two sets, it shows individuals speaking either or both languages, reflecting linguistic diversity.
  • The union encompasses all pets falling into either category, illustrating the variety of household animals.

Complement of Union of Sets in Venn Diagram

The complement of the union of sets in Venn Diagrams refers to elements not included in the union of specified sets, offering a unique perspective on set relationships. This concept is essential for educators teaching logical complementation, as it visually separates the universal set from the combined sets. By identifying elements outside the union, students gain insight into exclusion within set theory, enriching their understanding of mathematical and logical boundaries.

  • The complement shows students who participate in neither activity, highlighting diverse interests outside these areas.
  • This example identifies foods categorized outside of fruits and vegetables, emphasizing the variety in dietary choices.
  • The complement includes books outside these genres, showcasing the wide range of literature beyond specific categories.
  • By focusing on the complement, it reveals countries situated outside these continents, expanding geographic knowledge.
  • This complement helps in understanding the diversity of animal kingdoms beyond avian and aquatic life forms.

Complement of Intersection of Sets in Venn Diagram

Discover the concept of the complement of the intersection of sets in Venn Diagrams, a crucial element for students and teachers in understanding set theory. This principle highlights the elements that are not part of the intersection of two sets, offering a visual and intuitive method to grasp set relationships. By learning this concept, educators can enhance their teaching strategies, enabling students to better understand complex set operations through visual representation.

  • In a diagram of sets A and B, the complement of A ? B includes all elements outside the overlapping area. This illustrates elements not shared by A and B.
  • With the universal set U containing all possible elements, the complement of A ? B is represented by all areas in U not in A ? B , showcasing non-common elements.
  • In a Venn Diagram with sets A, B, and C, the complement of A ? B ? C shows elements in U excluding those common to all three sets, emphasizing unique elements outside these intersections.
  • Considering sets of ‘red items’ and ’round items’, the complement of their intersection excludes items that are both red and round, helping students identify items that do not possess both properties simultaneously.
  • This concept is used in probability to determine the likelihood of events not occurring together, enhancing students’ understanding of probability theory through visual aids.

What Is a Venn Diagram in Math?

A Venn Diagram is a visual tool used in math to show the relationships between different sets through overlapping circles.

How Do You Read a Venn Diagram?

To read a Venn Diagram, identify each circle as a set and where they overlap, indicating shared elements or characteristics.

Why Are They Called Venn Diagrams?

They are named after John Venn, a mathematician who popularized these diagrams in the 1880s to illustrate logical relationships.

What Is the Middle of a Venn Diagram Called?

The middle, where two or more circles overlap, is called the intersection, representing elements common to all overlapping sets.

Does a Venn Diagram Always Use 2 or 3 Circles?

No, a Venn Diagram can use any number of circles, though 2 or 3 are most common for simplicity and clarity in representation.

Venn Diagrams are indispensable tools in education, offering a visual representation of set relationships that enhance comprehension and analytical skills. By simplifying complex concepts, they become accessible, engaging, and insightful for students, making them a favored resource among educators dedicated to fostering a deeper understanding of mathematics and logic.

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What is a Venn Diagram

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Creating Venn diagrams is super simple and easy with our Venn diagram maker. Learn the essentials of Venn diagrams, along with their long history, versatile purposes and uses, examples and symbols, and steps to draw them. 

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What is a Venn diagram?

A Venn diagram uses overlapping circles or other shapes to illustrate the logical relationships between two or more sets of items. Often, they serve to graphically organize things, highlighting how the items are similar and different.

Venn diagrams, also called Set diagrams or Logic diagrams, are widely used in mathematics, statistics, logic, teaching, linguistics, computer science and business. Many people first encounter them in school as they study math or logic, since Venn diagrams became part of “new math” curricula in the 1960s. These may be simple diagrams involving two or three sets of a few elements, or they may become quite sophisticated, including 3D presentations, as they progress to six or seven sets and beyond. They are used to think through and depict how items relate to each within a particular “universe” or segment. Venn diagrams allow users to visualize data in clear, powerful ways, and therefore are commonly used in presentations and reports. They are closely related to Euler diagrams, which differ by omitting sets if no items exist in them. Venn diagrams show relationships even if a set is empty.

Venn diagram history

Venn diagrams are named after British logician John Venn. He wrote about them in an 1880 paper entitled “On the Diagrammatic and Mechanical Representation of Propositions and Reasonings” in the Philosophical Magazine and Journal of Science.

But the roots of this type of diagram go back much further, at least 600 years. In the 1200s, philosopher and logician Ramon Llull (sometimes spelled Lull) of Majorca used a similar type of diagram, wrote author M.E. Baron in a 1969 article tracing their history. She also credited German mathematician and philosopher Gottfried Wilhelm von Leibnitz with drawing similar diagrams in the late 1600s.

In the 1700s, Swiss mathematician Leonard Euler (pronounced Oy-ler) invented what came to be known as the Euler Diagram, the most direct forerunner of the Venn Diagram. In fact, John Venn referred to his own diagrams as Eulerian Circles, not Venn Diagrams. The term Venn Diagrams was first published by American philosopher Clarence Irving (C.I.) Lewis in his 1918 book, A Survey of Symbolic Logic.

Venn Diagrams continued to evolve over the past 60 years with advances by experts David W. Henderson, Peter Hamburger, Jerrold Griggs, Charles E. “Chip” Killian and Carla D. Savage.  Their work concerned symmetric Venn Diagrams and their relationship to prime numbers, or numbers indivisible by other numbers except 1 and the number itself. One such symmetric diagram, based on prime number 7, is widely known in math circles as Victoria.

Other notable names in the development of Venn Diagrams are A.W.F. Edwards, Branko  Grunbaum and Henry John Stephen Smith. Among other things, they changed the shapes in the diagrams to allow simpler depiction of Venn Diagrams at increasing numbers of sets.

Example Venn diagram

Say our universe is pets, and we want to compare which type of pet our family might agree on.

Set A contains my preferences: dog, bird, hamster.

Set B contains Family Member B’s preferences: dog, cat, fish.

Set C contains Family Member C’s preferences: dog, cat, turtle, snake.

The overlap, or intersection, of the three sets contains only dog. Looks like we’re getting a dog.

Of course, Venn diagrams can get a lot more involved than that, as they are used extensively in various fields.

Venn diagram purpose and benefits

  • To visually organize information to see the relationship between sets of items, such as commonalities and differences. Students and professionals can use them to think through the logic behind a concept and to depict the relationships for visual communication. This purpose can range from elementary to highly advanced.
  • To compare two or more choices and clearly see what they have in common versus what might distinguish them. This might be done for selecting an important product or service to buy.
  • To solve complex mathematical problems. Assuming you’re a mathematician, of course.
  • To compare data sets, find correlations and predict probabilities of certain occurrences.
  • To reason through the logic behind statements or equations, such as the Boolean logic behind a word search involving “or” and “and” statements and how they’re grouped.

Venn diagram use cases

  • Math: Venn diagrams are commonly used in school to teach basic math concepts such as sets, unions and intersections. They’re also used in advanced mathematics to solve complex problems and have been written about extensively in scholarly journals. Set theory is an entire branch of mathematics.
  • Statistics and probability: Statistics experts use Venn diagrams to predict the likelihood of certain occurrences. This ties in with the field of predictive analytics. Different data sets can be compared to find degrees of commonality and differences.
  • Logic: Venn diagrams are used to determine the validity of particular arguments and conclusions. In deductive reasoning, if the premises are true and the argument form is correct, then the conclusion must be true. For example, if all dogs are animals, and our pet Mojo is a dog, then Mojo has to be an animal. If we assign variables, then let’s say dogs are C, animals are A, and Mojo is B. In argument form, we say: All C are A. B is C. Therefore B is A. A related diagram in logic is called a Truth Table, which places the variables into columns to determine what is logically valid. Another related diagram is called the Randolph diagram, or R-Diagram, after mathematician John F. Randolph. It uses lines to define sets.
  • Linguistics: Venn diagrams have been used to study the commonalities and differences among languages.
  • Teaching reading comprehension: Teachers can use Venn diagrams to improve their students’ reading comprehension. Students can draw diagrams to compare and contrast ideas they are reading about.
  • Computer science: Programmers can use Venn diagrams to visualize computer languages and hierarchies.
  • Business: Venn diagrams can be used to compare and contrast products, services, processes or pretty much anything that can depicted in sets. And they’re an effective communication tool to illustrate that comparison.

Venn diagram glossary

On a lighter note: venn diagrams hit the small screen.

Not many diagrams have crossed over into popular culture, but the esteemed Venn diagram has.

  • Drama: In the CBS TV show NUMB3RS, produced from 2005 to 2010, math genius Charles Eppes uses a Venn diagram to determine which suspects match a description and have a history of violence.
  • Comedy: On NBC’s Late Night with Seth Meyers, the comedian has a recurring routine called “Venn Diagrams,” comparing two seemingly unrelated items to find their funny commonality (he hopes.)

Steps to draw and use a basic Venn diagram

  • Determine your goal. What are you comparing, and why? This will help you to define your sets.
  • Brainstorm and list the items in your sets, either on paper or with a platform like Lucidchart.
  • Now, use your diagram to compare and contrast the sets. You may see things in new ways and be able to make observations, choices, arguments or decisions.

Additional Resources

  • Venn Diagram Templates
  • How to Create a Venn Diagram in PowerPoint

Lucidchart lets you create professional-looking Venn diagrams with easy-to-use software. With all editing taking place in the cloud, it’s easy to collaborate with colleagues on a Venn diagram. You can even import images and share your diagram digitally or via print.

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Venn Diagram: Concept and Solved Questions

problem solving of venn diagram

What is a Venn Diagram?

Venn diagram, also known as Euler-Venn diagram is a simple representation of sets by diagrams. The usual depiction makes use of a rectangle as the universal set and circles for the sets under consideration.

In CAT and other MBA entrance exams, questions asked from this topic involve 2 or 3 variable only. Therefore, in this article we are going to discuss problems related to 2 and 3 variables.

Let's take a look at some basic formulas for Venn diagrams of two and three elements.

n ( A ∪ B) = n(A ) + n ( B ) - n ( A∩ B) n (A ∪ B ∪ C) = n(A ) + n ( B ) + n (C) - n ( A ∩ B) - n ( B ∩ C) - n ( C ∩ A) + n (A ∩ B ∩ C )

And so on, where n( A) = number of elements in set A.  Once you understand the concept of Venn diagram with the help of diagrams, you don’t have to memorize these formulas.

Venn Diagram in case of two elements

problem solving of venn diagram

Where;  X = number of elements that belong to set A only Y = number of elements that belong to set B only Z = number of elements that belong to set A and B both (AB) W = number of elements that belong to none of the sets A or B From the above figure, it is clear that  n(A) = x + z ;  n (B) = y + z ;  n(A ∩ B) = z; n ( A ∪ B) = x +y+ z. Total number of elements = x + y + z + w

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Venn Diagram in case of three elements

problem solving of venn diagram

Where, W = number of elements that belong to none of the sets A, B or C

Tip: Always start filling values in the Venn diagram from the innermost value.

Solved Examples

Example 1:  In a college, 200 students are randomly selected. 140 like tea, 120 like coffee and 80 like both tea and coffee. 

  • How many students like only tea?
  • How many students like only coffee?
  • How many students like neither tea nor coffee?
  • How many students like only one of tea or coffee?
  • How many students like at least one of the beverages?

Solution:  The given information may be represented by the following Venn diagram, where T = tea and C = coffee.

problem solving of venn diagram

  • Number of students who like only tea = 60
  • Number of students who like only coffee = 40
  • Number of students who like neither tea nor coffee = 20
  • Number of students who like only one of tea or coffee = 60 + 40 = 100
  • Number of students who like at least one of tea or coffee = n (only Tea) + n (only coffee) + n (both Tea & coffee) = 60 + 40 + 80 = 180

Example 2:  In a survey of 500 students of a college, it was found that 49% liked watching football, 53% liked watching hockey and 62% liked watching basketball. Also, 27% liked watching football and hockey both, 29% liked watching basketball and hockey both and 28% liked watching football and basket ball both. 5% liked watching none of these games.

  • How many students like watching all the three games?
  • Find the ratio of number of students who like watching only football to those who like watching only hockey.
  • Find the number of students who like watching only one of the three given games.
  • Find the number of students who like watching at least two of the given games.

Solution: n(F) = percentage of students who like watching football = 49% n(H) = percentage of students who like watching hockey = 53% n(B)= percentage of students who like watching basketball = 62% n ( F ∩ H) = 27% ; n (B ∩ H) = 29% ; n(F ∩ B) = 28% Since 5% like watching none of the given games so, n (F ∪ H ∪ B) = 95%. Now applying the basic formula, 95% = 49% + 53% + 62% -27% - 29% - 28% + n (F ∩ H ∩ B) Solving, you get n (F ∩ H ∩ B) = 15%.

Now, make the Venn diagram as per the information given. Note: All values in the Venn diagram are in percentage.

problem solving of venn diagram

  • Number of students who like watching all the three games = 15 % of 500 = 75.
  • Ratio of the number of students who like only football to those who like only hockey = (9% of 500)/(12% of 500) = 9/12 = 3:4.
  • The number of students who like watching only one of the three given games = (9% + 12% + 20%) of 500 = 205
  • The number of students who like watching at least two of the given games=(number of students who like watching only two of the games) +(number of students who like watching all the three games)= (12 + 13 + 14 + 15)% i.e. 54% of 500 = 270.

To know the importance of this topic, check out some previous year CAT questions from this topic:

CAT 2017 Solved Questions:

Solution:  It is given that 200 candidates scored above 90th percentile overall in CET. Let the following Venn diagram represent the number of persons who scored above 80 percentile in CET in each of the three sections:

problem solving of venn diagram

2.  From the given condition, g is a multiple of 5. Hence, g = 20. The number of candidates at or above 90th percentile overall and at or above 80th percentile in both P and M = e + g = 60.

3.  In this case, g = 20. Number of candidates shortlisted for AET = d + e + f + g = 10 + 40 + 100 + 20 = 170

4.  From the given condition, the number of candidates at or above 90th percentile overall and at or above 80th percentile in P in CET = 104. The number of candidates who have to sit for separate test = 296 + 3 = 299.

Another type of questions asked from this topic is based on maxima and minima. We have discussed this type in the other article.

Key Learning:

  • It is important to carefully list the conditions given in the question in the form of a Venn diagram.
  • While solving such questions, avoid taking many variables.
  • Try solving the questions using the Venn diagram approach and not with the help of formulae.

You can also post in the comment section below, any query or explanation for any concept mentioned in the article.

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Venn diagrams

Venn diagrams are the diagrams that are used to represent the sets, relation between the sets and operation performed on them, in a pictorial way. Venn diagram, introduced by John Venn (1834-1883), uses circles (overlapping, intersecting and non-intersecting), to denote the relationship between sets. A Venn diagram is also called a set diagram or a logic diagram showing different set operations such as the intersection of sets,  union of sets and difference of sets. It is also used to depict subsets of a set.

For example, a set of natural numbers is a subset of whole numbers, which is a subset of integers. The relation between the sets of natural numbers, whole numbers and integers can be shown by the Venn diagram, where the set of integers is the universal set .  See the figure below.

Venn Diagram

Here, W represents whole numbers and N represents natural numbers

The universal set (U) is usually represented by a closed rectangle, consisting of all the sets. The sets and subsets are shown by using circles or oval shapes.

What is a Venn Diagram?

A diagram used to represent all possible relations of different sets. A Venn diagram can be represented by any closed figure, whether it be a Circle or a Polygon (square, hexagon, etc.). But usually, we use circles to represent each set. 

Venn diagram X and Y

In the above figure, we can see a Venn diagram, represented by a rectangular shape about the universal set, which has two independent sets, X and Y. Therefore, X and Y are disjoint sets. The two sets, X and Y, are represented in a circular shape. This diagram shows that set X and set Y have no relation between each other, but they are a part of a universal set.

For example, set X = {Set of even numbers} and set Y = {Set of odd numbers} and Universal set, U = {set of natural numbers}

We can use the below formula to solve the problems based on two sets.

n(X ⋃ Y) = n(X) + n(Y) – n(X ⋂ Y)

Venn Diagram of Three Sets

Venn diagram 3 sets

The formula used to solve the problems on Venn diagrams with three sets is given below:

n(A ⋃ B ⋃ C) = n(A) + n(B) + n(C) – n(A ⋂ B) – n(B ⋂ C) – n(A ⋂ C) + n(A ⋂ B ⋂ C)

Venn Diagram Symbols

The symbols used while representing the operations of sets are:

  • Union of sets symbol: ∪
  • Intersection of sets symbol: ∩
  • Complement of set: A’ or A c

How to draw a Venn diagram?

To draw a Venn diagram, first, the universal set should be known. Now, every set is the subset of the universal set (U). This means that every other set will be inside the rectangle which represents the universal set.

So, any set A (shaded region) will be represented as follows:

Venn Diagram Construction

Where U is a universal set.

We can say from fig. 1 that

All the elements of set A are inside the circle. Also, they are part of the big rectangle which makes them the elements of set U.

Venn Diagrams of  Set operations

In set theory, there are many operations performed on sets, such as:

  • Union of Set
  • Intersection of set
  • Complement of set
  • Difference of set

etc. The representations of different operations on a set are as follows:

Complement of a set in Venn Diagram

A’ is the complement of set A (represented by the shaded region in fig. 2). This set contains all the elements which are not there in set A.

Complement of set A

It is clear that from the above figure,

A + A’ = U 

It means that the set formed with elements of set A and set A’ combined is equal to U.

The complement of a complement set is a set itself.

Properties of Complement of set:

  • (A ∪ B)′ = A′ ∩ B′
  • (A ∩ B)′ = A′ ∪ B′

Intersection of two sets in Venn Diagram

A intersection B is given by: A ∩ B = {x : x ∈ A and x ∈ B}.

This represents the common elements between set A and B (represented by the shaded region in fig. 3).

A intersection B

Intersection of two Sets

Properties of the intersection of sets operation:

  • A ∩ B = B ∩ A
  • (A ∩ B) ∩ C = A ∩ (B ∩ C)
  • φ ∩ A = φ ; U ∩ A = A
  • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
  • A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Union of Two Sets in Venn Diagram

A union B is given by:  A ∪ B = {x | x ∈A or x ∈B} .

This represents the combined elements of set A and B (represented by the shaded region in fig. 4).

A union B

Union of two sets

Some properties of Union operation:

  • A ∪ B = B ∪ A
  • (A ∪ B) ∪ C = A ∪ (B ∪ C)

Complement of Union of Sets in Venn Diagram

(A ∪ B)’ : This is read as complement of A union B . This represents elements which are neither in set A nor in set B (represented by the shaded region in fig. 5).

Complement of Sets A union B

Complement of A U B

Complement of Intersection of Sets in Venn Diagram

(A ∩ B)’: This is read as complement of A intersection B . This represents elements of the universal set which are not common between set A and B (represented by the shaded region in fig. 6).

Complement of Intersection of Sets A and B

Complement of A ∩ B

Difference between Two Sets in Venn Diagram

A – B : This is read as A difference B . Sometimes, it is also referred to as ‘ relative complement ’. This represents elements of set A which are not there in set B(represented by the shaded region in fig. 7).

A Difference B

Difference between Two Sets

Symmetric difference between two sets in Venn Diagram

A ⊝ B: This is read as a  symmetric difference of set A and B . This is a set which contains the elements which are either in set A or in set B but not in both (represented by the shaded region in fig. 8).

Symmetric difference of A and B

Symmetric difference between two sets

Related Articles

  • Set Theory in Maths
  • Set Operations
  • Subset And Superset
  • Intersection And Difference Of Two Sets

Venn Diagram Example

Example: In a class of 50 students, 10 take Guitar lessons and 20 take singing classes, and 4 take both. Find the number of students who don’t take either Guitar or singing lessons.

Venn diagram example

Let A = no. of students who take guitar lessons = 10.

Let B = no. of students who take singing lessons = 20.

Let C = no. of students who take both = 4.

Now we subtract the value of C from both A and B. Let the new values be stored in D and E.

D = 10 – 4 = 6

E = 20 – 4 = 16

Now logic dictates that if we add the values of C, D, E and the unknown quantity “X”, we should get a total of 50 right? That’s correct.

So the final answer is X = 50 – C – D – E

X = 50 – 4 – 6 – 16

Venn’s diagrams are particularly helpful in solving word problems on number operations that involve counting. Once it is drawn for a given problem, the rest should be a piece of cake.

Venn Diagram Questions

  • Out of 120 students in a school, 5% can play Cricket, Chess and Carroms. If so happens that the number of players who can play any and only two games are 30. The number of students who can play Cricket alone is 40. What is the total number of those who can play Chess alone or Carroms alone?
  • Draw the diagram that best represents the relationship among the given classes: Animal, Tiger, Vehicle, Car
  • At an overpriced department store, there are 112 customers. If 43 have purchased shirts, 57 have purchased pants, and 38 have purchased neither, how many purchased both shirts and pants?
  • In a group, 25 people like tea or coffee; of these, 15 like tea and 6 like coffee and tea. How many like coffee?

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IMAGES

  1. Solving Problem using Venn Diagram Part 1

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  2. Problem Solving of Sets with 2 circles Venn Diagram

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  3. How To Solve Venn Diagram

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  4. Problem Solving with Venn Diagrams

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  5. Math 7 Week 2

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  6. Solving Word Problems With Venn Diagrams Three Sets

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VIDEO

  1. Introduction to Venn diagrams (مقدمة لشكل فين)

  2. Problem Solving Involving Venn Diagram

  3. Problem Solving in Venn Diagrams

  4. PROBLEM SOLVING INVOLVING SETS AND VENN DIAGRAM PART 2| @LoveMATHTV

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  6. Problems Involving Sets │Using Venn Diagrams │Math 7

COMMENTS

  1. 15 Venn Diagram Questions And Practice Problems With Solutions

    How to solve Venn diagram questions In middle school, sets and set notation are introduced when working with Venn diagrams. A set is a collection of objects. We identify a set using braces. For example, if set A contains the odd numbers between 1 and 10, then we can write this as: A = {1, 3, 5, 7, 9}

  2. Venn Diagram Examples, Problems and Solutions

    Definition and meaning. Venn diagram formula with an explanation. Examples of 2 and 3 sets Venn diagrams: practice problems with solutions, questions, and answers. Simple 4 circles Venn diagram with word problems. Compare and contrast Venn diagram example. Let's define it:

  3. Venn Diagrams: Exercises

    Venn diagram word problems generally give you two or three classifications and a bunch of numbers. You then have to use the given information to populate the diagram and figure out the remaining information. For instance: Out of forty students, 14 are taking English Composition and 29 are taking Chemistry.

  4. Venn Diagram Word Problems

    How To Use Venn Diagrams To Help Solve Counting Word Problems?

  5. Art of Problem Solving: Venn Diagrams with Two Categories

    Created using YouTube Video Editor Art of Problem Solving's Richard Rusczyk introduces 2-circle Venn diagrams, and using subtraction as a counting technique.Learn more:...

  6. How to Solve Problems Using Venn Diagrams

    Step 1: Understand the Problem As with any problem-solving method, the first step is to understand the problem. What sets are involved? How are they related? What are you being asked to find? Step 2: Draw the Diagram Draw a rectangle to represent the universal set, which includes all possible elements.

  7. Venn Diagram

    A Venn diagram is a diagram that helps us visualize the logical relationship between sets and their elements and helps us solve examples based on these sets. A Venn diagram typically uses intersecting and non-intersecting circles (although other closed figures like squares may be used) to denote the relationship between sets. Venn Diagram Example

  8. 1.3 Understanding Venn Diagrams

    To create a Venn diagram, first we draw a rectangle and label the universal set " U = Plants. " Then we draw a circle within the universal set and label it with the word "Trees.". Figure 1.7. This section will introduce how to interpret and construct Venn diagrams. In future sections, as we expand our knowledge of relationships between ...

  9. Venn diagram

    The following diagram is a Venn diagram for sets and : The red region contains all the elements that are in only. The blue region contains all the elements that are in only. The black region contains all the elements in both and which is called the intersection of and , denoted . The red, black, and blue regions together represent the elements ...

  10. Venn Diagram Questions With Solution

    Solution: Given, n (A) = 24, n (B) = 22 and n (A ∩ B) = 8 The Venn diagram for the given information is: (i) n (A ∪ B) = n (A) + n (B) - n (A ∩ B) = 24 + 22 - 8 = 38. (ii) n (A - B) = n (A) - n (A ∩ B) = 24 - 8 = 16. (iii) n (B - A) = n (B) - n (A ∩ B) = 22 - 8 = 14.

  11. Sets and Venn Diagrams

    T means the set of Tennis players. V means the set of Volleyball players. The Venn Diagram is now like this: Union of 3 Sets: S ∪ T ∪ V. You can see (for example) that: drew plays Soccer, Tennis and Volleyball. jade plays Tennis and Volleyball. alex and hunter play Soccer, but don't play Tennis or Volleyball. no-one plays only Tennis.

  12. 2.2: Venn Diagrams

    2.2: Venn Diagrams. Page ID. Julie Harland. MiraCosta College. This is a Venn diagram using only one set, A. This is a Venn diagram Below using two sets, A and B. This is a Venn diagram using sets A, B and C. Study the Venn diagrams on this and the following pages. It takes a whole lot of practice to shade or identify regions of Venn diagrams.

  13. Venn Diagram Examples for Problem Solving

    Venn Diagram Examples for Problem Solving Updated on: 13 September 2022 What is a Venn Diagram? Venn diagrams define all the possible relationships between collections of sets. The most basic Venn diagrams simply consist of multiple circular boundaries describing the range of sets. A simple Venn diagram example

  14. Solving Problems with Venn Diagrams

    Solving Problems with Venn Diagrams Mathispower4u 277K subscribers Subscribe Subscribed 6.5K 908K views 12 years ago Sets This video solves two problems using Venn Diagrams. One with...

  15. Venn diagrams

    Solving problems using Venn diagrams You may be asked to solve problems using Venn diagrams in an exam. It is really important you draw the Venn diagram and add information as you go...

  16. Art of Problem Solving: Venn Diagrams with Three Categories

    Art of Problem Solving's Richard Rusczyk introduces 3-circle Venn diagrams as a counting technique.Learn more about problem solving here: http://bit.ly/Artof...

  17. PDF Set Theory: Venn Diagrams for Problem Solving

    Solution Create a Venn diagram with two sets. To do this, first draw two intersecting circles inside a rectangle. Then, make sure to work from the inside out. That is, first place the 60 customers that fall into the intersection of the two sets.

  18. PDF Solving Problems using Venn Diagrams LESSON

    Solving Problems using Venn Diagrams Starter 1. The Venn diagram alongside shows the number of people in a sporting club who play tennis (T) and hockey (H). Find the number of people: (a) in the club (b) who play hockey (c) who play both sports (d) who play neither sport (e) who play at least one sport (f) who play tennis but not hockey T H

  19. Venn Diagrams

    Venn Diagrams serve as versatile tools in education, data analysis, and problem-solving. By visually mapping out relationships between sets, they facilitate understanding of complex concepts through overlap and distinction.

  20. What is a Venn Diagram

    The Ultimate Venn Diagram Guide - Includes the history of Venn diagrams, benefits to using them, examples, and use cases. Learn about terminology and how to draw a basic Venn diagram. ... They're also used in advanced mathematics to solve complex problems and have been written about extensively in scholarly journals. Set theory is an entire ...

  21. Problem solving Venn Diagrams- 3 sets HL

    This video explains how to fill out a Venn diagram given key information, including examples using algebra.ExamRevision is Ireland's leading video tutorial w...

  22. Venn Diagram

    Let's take a look at some basic formulas for Venn diagrams of two and three elements. n ( A ∪ B) = n (A ) + n ( B ) - n ( A∩ B) n (A ∪ B ∪ C) = n (A ) + n ( B ) + n (C) - n ( A ∩ B) - n ( B ∩ C) - n ( C ∩ A) + n (A ∩ B ∩ C ) And so on, where n ( A) = number of elements in set A.

  23. Venn Diagrams of Sets

    A Venn diagram is also called a set diagram or a logic diagram showing different set operations such as the intersection of sets, union of sets and difference of sets. It is also used to depict subsets of a set. For example, a set of natural numbers is a subset of whole numbers, which is a subset of integers.

  24. Problem Solving with Venn diagrams

    Solve worded problems using Venn diagrams.