problem solving venn diagrams 2 circles

VENN DIAGRAM WORD PROBLEMS WITH 2 CIRCLES

Example 1 :

In a class of 50 students, each of the students passed either in mathematics or in  science or in both. 10 students passed in both and 28 passed in science. Find how many  students passed in mathematics?

Let M = The set of students passed in Mathematics

S = The set of students passed in Science

We may solve the given problem using two methods.

(i) Using formula

(ii) Using venn diagram

Method 1 : 

Total number of students   n (M U S)   =  50

Number of students passed in both subjects n(MnS) = 10

Number of students passed in science n (S)  =  28

From this, we have to find the number of students who passed in mathematics.

n (M U S)  = n (M) + n (S) -  n (M n S)

50  =  n (M) + 28 - 10

50  =  n (M) + 18

Subtract 18 on both sides

50 - 18 =  n (M) + 18 - 18

So, the number students passed in Mathematics is 32.

Let "x" be the number of students passed in Mathematics.

By representing the given details in venn diagram, we get

From the Venn diagram

x + 10 + 18 = 50

x = 50 - 28 = 22

Number of students passed in Mathematics

= x + 10 = 22 + 10 = 32

Example 2 :

The population of a town is 10000. Out of these 5400 persons read newspaper A and  4700 read newspaper B. 1500 persons read both the newspapers. Find the number of  persons who do not read either   of the two papers.

Let A = The set of persons who read newspaper A

B = The set of persons who read  newspaper B

Number of persons who read at least one news paper

= 3900 + 1500 + 3200

=  8600

Total population  =  10000

To find the number of  persons who do not read either   of the two papers, we have to subtract number of persons who read at least one from total population.

  =  10000 - 8600

  =  1400

So, the number of  persons who do not read either   of the two papers is 1400.

Example 3 :

In a school, all the students play either Foot ball or Volley ball or both. 300 students  play Foot ball, 270 students play Volley ball and 120 students play both games. Find

(i) the number of students who play Foot ball only

(ii) the number of students who play Volley ball only

(iii) the total number of students in the school

Let A = The set of students who play foot ball 

B = The set of students who play volley ball

(i) The number of students who play Foot ball only is 1 80

(ii)The number of students who play Volley ball only is 150

(iii) The total number of students in the school

  =  180 + 120 + 150

  =  450

Example 4 :

In a School 150 students passed X Standard Examination. 95 students applied for  Group I and 82 students applied for Group II in the Higher Secondary course. If 20  students applied neither of the two, how many students applied for both groups?

A = The set of students who applied for Group I

B = The set of students who applied for Group II

Number of students who applied at least one group

  =  150 - 20

  =  130

n (A) = 95, n (B) = 82 and n (A U B) = 130

n (A U B) = n (A) + n (B) - n (A n B)

130 =  95 + 82  - n (A n B)

130  =  177 -  n (A n B)

 n (A n B)  =  177 - 130  ==> 47

So, the number of students  applied for both groups is 47.

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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.

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Venn diagram word problems

The Venn diagram word problems in this lesson will show you how to use Venn diagrams with 2 circles to solve problems involving counting. 

Venn diagram word problems with two circles

Word problem #1

A survey was conducted in a neighborhood with 128 families. The survey revealed the following information.

• 106 of the families have a credit card
• 73 of the families are trying to pay off a car loan
• 61 of the families have both a credit card and a car loan

Answer the following questions:

1. How many families have only a credit card?

2.  How many families have only a car loan?

3. How many families have neither a credit card nor a car loan?

4. How many families do not have a credit card?

5. How many families do not have a car loan?

6. How many families have a credit card or a car loan?

• Let C be families with a credit card
• Let L be families with a car loan
• Let S be the total number of families

The Venn diagram above can be used to answer all these questions. 

Tips on how to create the Venn diagram. Always put  first , in the middle or in the intersection, the value that is in both sets. For example, since 61 families have both a credit card and a car loan, put 61 in the intersection before you do anything else. In C only, put 45 since 106 - 61 = 45

In L only, put 12 since 73 - 61 = 12

Outside C and L, put 10 since 128 - 61 - 45 - 12 = 10

The expression, " only a credit card" means that it is only in C. Any number in L cannot be included. 1.  The number of families with only a credit card is 45. Do not add 61 to 45 since 61 is in L.

2.  The number of families with only a car loan is 12. 

3. The number of families with neither a credit card nor a car loan is 10. 10 is not in C nor in L.

4. The number families without a credit card is found by adding everything that is not in C. 12 + 10 = 22

5.  The number families without a car loan is found by adding everything that is not in L. 45 + 10 = 55

6. The number of families with a credit card or a car loan is found by adding anything in C only, in L only and in the intersection of C and L?

45 + 61 + 12 = 118

Word problem #2

A survey conducted in a school with 150 students revealed the following information:

• 78 students are enrolled in swimming class
• 85 students are enrolled in basketball class
• 25 are enrolled in both swimming and basketball class

1.  How many students are enrolled only in swimming class?

2.  How many students are enrolled only in basketball class?

3.  How many students are neither enrolled in swimming class nor basketball class?

4.  How many students are not enrolled in swimming class?

5.  How many students are not enrolled in basketball class?

6. How many students are enrolled in swimming class or basketball class?

• Let S be students enrolled in swimming class
• Let B be students enrolled in basketball class
• Let E be the total number of students

Using the same technique as in problem #1 , we have the following Venn diagram

1. The number of students enrolled only in swimming class is 53 2.  The number of students enrolled only in basketball class is 60

3. The number of students who are neither enrolled in swimming class nor basketball class is 12

4. Students not enrolled in swimming class are enrolled in basketball class only or are enrolled in neither of these two activities. In other words, everything that is not in S.

60 + 12 = 72

5.  Students not enrolled in basketball class are enrolled in swimming class only or are  enrolled in neither of these two activities. In other words, everything that is not in B.

53 + 12 = 65

6.  The number of students enrolled in swimming class or basketball class is found by adding anything in S only, in B only and in the intersection of S and B?

53 + 25 + 60 = 138

A tricky Venn diagram word problem with two circles

Word problem #3

In a survey of 100 people, 28 people smoke, 65 people drink, and 30 people do neither. How many people do both?

• Let K be the number of people who smoke
• Let D be  the number of people who drink
• Let E be the total number of people
• Let x be the number of people who smoke and drink

If we make a Venn diagram, here is what we have so far.

We end up with the following equation to solve for x.

(65 - x) + x + (28 - x) + 30 = 100

65 - x + x + 28 - x + 30 - 30 = 100 - 30

65 - x + x + 28 - x  = 70

65 + 0 + 28 - x  = 70

93 - x  = 70

Since 93 - 23 = 70, x  = 23

The number of people who do both is 23.

3-circle Venn diagram

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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?

Content Continues Below

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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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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.

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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• Word Problems: Two sets

Venn Diagram Word Problem Worksheets: Two sets

Venn diagram word problems are based on union, intersection, complement and difference of two sets. This batch of printable word problems on Venn diagram with two circles for students of grade 5 through grade 8 is illustrated with images, numbers, words and symbols. Few word problems may contain universal set. Use the key information from the given data to solve the word problems. Click on our free worksheets and kick-start your practice!

Reading Venn Diagram - Type 1

The elements of the set are represented as pictures on the two circles of these Venn diagrams. Read each Venn diagram and answer the given questions.

• Download the set

Reading Venn Diagram - Type 2

The elements of the set are multiples, names, days, etc. Answer the word problems that follow the Venn diagram on each printable worksheet. Children should learn to differentiate the overlapping regions and the relation between the sets at the end of the practice.

Reading Venn Diagram - Type 3

These Venn diagram word problem pdfs require the 5th grade and 6th grade children to count the number of elements in the region. The cross-markings represent the customers in cafes, pizzeria, books in the library, park goers, or members of a gym.

Venn Diagram Word Problems - No Universal Set

Several practice problems on 2-circle Venn diagrams without universal set are given in these printable worksheets. Read each Venn diagram carefully and write down the answer.

Venn Diagram Word Problems - With a Universal Set

These Venn diagram word problems worksheet pdfs feature two sets representing the quantities of the data. Analyze the regions including universal set to solve these word problems.

Drawing Venn Diagram - No Universal Set

Follow the direction and create a Venn diagram with two intersecting circles for the given data. Fill in the key information to complete the Venn diagram with 2 circles. Use all the available information to answer the questions.

Drawing Venn Diagram - With a Universal Set

Task 7th grade and 8th grade students to draw a Venn diagram with a universal set and fill in each region with the information provided. Interpret the Venn diagram and answer the word problems given below.

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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 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.

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.

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.

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.

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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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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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

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.

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 Worksheet

Download this quiz to check your students' understanding of Venn diagrams. Includes 10 questions with answers!

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.

Venn diagram questions 6th grade

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

How many people have brown hair and glasses?

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?

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.

Venn diagram questions 7th grade

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.

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?

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

Venn diagram questions 8th grade

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

\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.

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.

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.

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.

Lower ability Venn diagram questions

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.

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.

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

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}).

\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.

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

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

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

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

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

Middle ability high school Venn diagram questions

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.

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.

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

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}.

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

H = history

G = geography

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.

Put this information onto a Venn diagram.

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

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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 Word Problems

Venn Diagram Word Problems can be very easy to make mistakes on when you are a beginner.

It is extremely important to:

Read the question carefully and note down all key information.

Know the standard parts of a Venn Diagram

Work in a step by step manner

Check at the end that all the numbers add up coorectly.

Let’s start with an easy example of a two circle diagram problem.

Venn Diagrams – Word Problem One

“A class of 28 students were surveyed and asked if they ever had dogs or cats for pets at home. 8 students said they had only ever had a dog. 6 students said they had only ever had a cat. 10 students said they had a dog and a cat. 4 students said they had never had a dog or a cat.”

Note that the word “only” is extremely important in Venn Diagram word problems.

Because the word “only” is in our problem text, it makes it an easy word problem. Since this question is about dogs and cats, it will require a two circle Venn Diagram.

Here is the type of diagram we will need.

Our problem is an easy one where we have been given all of the numbers for the items required on the diagram.

We do not need to work out any missing values.

All we need to do is place the numbers from the word problem onto the standard Venn Diagram and we are done.

Venn Diagrams – Word Problem Two

The answer for this question will actually be the same as the Cats and Dogs question in Example 1.

However this time we are given less information, and so we will have work out the missing information.

Here is Problem 2: “A class of 28 students were surveyed and asked if they ever had dogs or cats for pets at home. 18 students said they had a dog. 16 students said they had a cat. 4 students said they had never had a dog or a cat.”

The above question does not contain the word “only” anywhere in it, and this is an indication that we will have to do some working out.

The question states that: “18 students said they had a dog” without the word “only” in there.

This means that the total of the Dogs circle is 18.

The 18 total students for Dogs includes people that have both a cat and a dog, as well as people who only have a dog.

Some people, who do not read this question carefully, will simply take the above figures and put them straight into a Venn Diagram like this.

Always check at the end that the numbers add up to the “E” Grand total.

16 + 18 + 4 = 38 which is much bigger than the “E” total of 28.

From the given information we have been able to work out that the circles total is 24. (Eg. Everything Total – No Cats and No Dogs = 28 – 4 = 24. This is vital information we now use to work on the rest of the problem.

Let’s first work out the “Only Cats” value.

Next we work out the “Only Dogs” number of people.

All we have left to work out is the number of Cats and Dogs for the center of the diagram.

We can do this any of three possible ways: Cats and Dogs = Total Cats – Only Cats or

Cats and Dogs = Total Dogs – Only Dogs

Cats and Dogs = E Total – Only Cats – Only Dogs – (No cats and No Dogs)

Any way that we work it out, the answer is 10.

So here is the final completed Venn Diagram Answer.

When putting answers into our Mathematics Workbook, we do not have to color in the diagram.

A final answer like the following is quite acceptable.

We can summarise the steps we used to work out this problem as follows.

Word Problem Two – Summary of Steps

– Work out What Information is given, and what needs to be calculated.

– Circles Total = E everything – (No Cats and No Dogs) – Cats Only = Circles Total – Total Dogs

– Dogs Only = Circles Total – Total Cats

– Cats and Dogs = Cats Total – Cats Only

– Finally, check that all the numbers in the diagram add up to equal the “E” everything total.

Word Problem Three – Subsets

“Fifty people were surveyed and only 20 people said that they regularly eat Healthy Foods like Fruit and Vegetables. Of these 20 healthy eaters, 12 said that they ate Vegetables every day. Draw a Venn Diagram to represent these results.”

This problem is quite different to our other two circle diagrams.

Cats and Dogs are very different to each other, and so we needed two separate circles.

However Healthy Foods and Vegetables are not different to each other because Vegetables are a type of Healthy Food.

We say that vegetables are a “Subset” of Healthy Foods.

This means that we do not separate the circles. We actually need to draw our circles inside each other like this.

The total adds up to 50, and the 12 people who include vegetables in their healthy foods are shown as being fully inside the Healthy Foods circle.

Word Problem Four – Disjoint Sets “Draw a Venn Diagram which divides the twelve months of the year into the following two groups: Months whose name begins with the letter “J” and Months whose name ends in “ber”. You will need a two circle Venn Diagram for your answer.” The first step is to list the twelve months of the year:

January – named after Janus, the god of doors and gates February – named after Februalia, when sacrifices were made for sins March – named after Mars, the god of war April – from aperire, Latin for “to open” (buds) May – named after Maia, the goddess of growth of plants June – named after junius, Latin for the goddess Juno July – named after Julius Caesar in 44 B.C. August – named after Augustus Caesar in 8 B.C. September – from septem, Latin for “seven” October – from octo, Latin for “eight” November – from novem, Latin for “nine” December – from decem, Latin for “ten”

Months starting with J = { January, June, July }

Months ending in “ber” = { September, October, November, December }

The two sets do not have any items in common, and so we will not overlap them. The remaining months will need to go outside of our two circles.

There should be all twelve months in the diagram when we are finished.

The completed Venn Diagram is shown below:

Venn Word Problems – Summary We have not included three circle diagrams, as they will be covered in a separate lesson.

Remember the working out steps for harder problems are:

Work out What Information is given, and what needs to be calculated.

Check to see if the two sets are “Subsets” or “Disjoint” sets.

If they are “Intersecting Sets” then some of the following formulas may be needed.

Circles Total = E everything – (Not in A and Not in B)

In A Only = Both Circles Total – Total in B

In A Only = The A Circle Total – Total in the intersection (A and B)

In B Only = Both Circles Total – Total in A

In B Only = The B Circle Total – Total in the intersection (A and B)

In the Intersection (A and B) = Total in B – In B Only

In the Intersection (A and B) = Total in A – In A Only

Finally, check that the numbers in the diagram all add up to equal the “E” everything total.

Venn Word Problems Videos

The following video shows a typical two circles word problem.

Here is a video that covers a two circles problem, where we need to find the number of items that are ( not in “A” and not in “B”)

Here is a Video which shows how to solve Venn Diagram Survey Problems.

Related Items

Introduction to Venn Diagrams Three Circle Venn Diagrams Real World Venn Diagrams

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1.3 Understanding Venn Diagrams

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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2.2: Venn Diagrams

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• Julie Harland
• MiraCosta College

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.

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.

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

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

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

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 \(B \cup C\)

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 .

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 )

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.

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.

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

Exercise 14

( A \(\cap\) C ) – B

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

Exercise 19

Exercise 20, exercise 21, exercise 22.

A \(\cap\) B

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!!

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

Venn Diagram Calculator

More options are available after generating the Venn Diagram, including the ability to change the opacity of the filling and the colors of the borders.

Venn diagram online

Venn diagram example, how to use the venn diagram calculator:, 1. data list:.

• In this case, the " Output " field will appear, and you can choose one of the following options:

2. Number of data items:

• For groups with intersections, you can enter two numbers:

How to solve a Venn diagram problem.

When problem contains numbers of items, when problem contains data lists, calculators.

• 1-800-234-2933
• [email protected]

Venn Diagram (2 circles) Calculator

Calculate all items of the Venn Diagram above

A = B = C =

Calculate P(A):

P(A) = Only Items in the A circle (no sharing) = = 0

Calculate P(B):

P(B) = Only Items in the B circle (no sharing) = = 0

Calculate P(A U B):

P(A U B) = A + B

P(A U B) = 0 + 0 = 0

Calculate P(A ∩ B):

P(A ∩ B) = Only Items that A and B share = = 0

Calculate P(A c ):

P(A c ) = Everything outside the A circle

P(A c ) = 1 - P(A)

P(A c ) = 1 - 0

P(A c ) = 1

Calculate P(B c ):

P(B c ) = Everything outside the B circle

P(B c ) = 1 - P(B)

P(B c ) = 1 - 0

P(B c ) = 1

Calculate P(A U B) c :

P(A U B) c = Only Items outside of A ∩ B

P(A U B) c = 1 - P(A U B)

P(A U B) c = 1 - 0

P(A U B) c = 1

Calculate P(A ∩ B) c :

P(A ∩ B) c = Only Items outside of A ∩ B

P(A ∩ B) c = 1 - P(A ∩ B)

P(A ∩ B) c = 1 - 0

P(A ∩ B) c = 1

Calculate P(A c ∩ B c )

P(A c ∩ B c ) = 1 - P(A) - P(B) + P(A ∩ B)

P(A c ∩ B c ) = 1 - 0 - 0 + 0

P(A c ∩ B c ) = 1 - 0

You have 2 free calculationss remaining

What is the answer, how does the venn diagram (2 circles) calculator work, what 1 formula is used for the venn diagram (2 circles) calculator, what 4 concepts are covered in the venn diagram (2 circles) calculator.

• intersection
• probability
• venn diagram (2 circles)
• venn diagram

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