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How to Create a Venn Diagram in Microsoft PowerPoint
Marshall is a writer with experience in the data storage industry. He worked at Synology, and most recently as CMO and technical staff writer at StorageReview. He's currently an API/Software Technical Writer based in Tokyo, Japan, runs VGKAMI and ITEnterpriser, and spends what little free time he has learning Japanese. Read more...
One of Microsoft PowerPoint’s charms is the ability to convey messages through illustrations, images, and SmartArt graphics. In its library of SmartArt graphics, PowerPoint provides a Venn diagram template, which you can completely customize to fit your needs.
Insert a Venn Diagram
Open PowerPoint and navigate to the “Insert” tab. Here, click “SmartArt” in the “Illustrations” group.
The “Choose A SmartArt Graphic” window will appear. In the left-hand pane, select “Relationship.”
Next, choose “Basic Venn” from the group of options that appear. Once selected, a preview and a description of the graphic will appear in the right-hand pane. Select the “OK” button to insert the graphic.
Once inserted, you can customize the Venn diagram.
RELATED: How to Insert a Picture or Other Object in Microsoft Office
Customize Your Venn Diagram
There are different ways you can customize your Venn diagram. For starters, you probably want to adjust the size . To do so, click and drag the corner of the SmartArt box. You can also resize individual circles within the diagram by selecting the circle and dragging the corner of its box.
Once resized, you can edit the text in each circle by clicking the circle and typing in the text box. Alternatively, you can click the arrow that appears at the left of the SmartArt box and then enter your text in each bullet.
To add additional circles to the diagram, just click “Enter” in the content box to add another bullet point. Similarly, removing a bullet point will remove that circle from the diagram.
To add text where the circles overlap, you’ll need to manually add a text box and enter text. To add a text box, select “Text Box” in the “Text” group of the “Insert” tab.
You’ll now notice your cursor changes to a down arrow. Click and drag to draw your text box, and then enter text.
Repeat this step until you’ve added all the text required for your Venn diagram.
PowerPoint also offers a few color variations for the SmartArt graphic. Select the SmartArt and then click the “Design” tab that appears. Here, choose “Change Colors” in the “SmartArt Styles” group.
Select the color scheme you like from the drop-down menu that appears.
You can also change the color of individual circles by right-clicking the border of the circle and selecting “Format Shape” from the context menu.
The “Format Shape” pane will appear in the right-hand side of the window. In the “Shape Options” tab, click “Fill” to display its options, click the box next to “Color,” then select your color from the palette.
Repeat this process for each circle in the diagram until you’re satisfied with the color scheme of your Venn diagram.
Assigning different colors to each circle in the diagram can make the relationship between subjects more distinct.
RELATED: How to Create a Timeline in Microsoft PowerPoint
- › How to Create and Insert a Pyramid in Microsoft PowerPoint
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Venn Diagrams to Plan Essays and More
- M.Ed., Education Administration, University of Georgia
- B.A., History, Armstrong State University
A Venn diagram is a great tool for brainstorming and creating a comparison between two or more objects, events, or people. You can use this as a first step to creating an outline for a compare and contrast essay .
Simply draw two (or three) large circles and give each circle a title, reflecting each object, trait, or person you are comparing.
Inside the intersection of the two circles (overlapping area), write all the traits that the objects have in common. You will refer to these traits when you compare similar characteristics.
In the areas outside the overlapping section, you will write all of the traits that are specific to that particular object or person.
Creating an Outline for Your Essay Using a Venn Diagram
From the Venn diagram above, you can create an easy outline for your paper. Here is the beginning of an essay outline:
1. Both dogs and cats make great pets.
- Both animals can be very entertaining
- Each is loving in its own way
- Each can live inside or outside the house
2. Both have drawbacks, as well.
- They can damage property
- Both can be costly
- Both require time and attention
3. Cats can be easier to care for.
- Leaving for a day
4. Dogs can be better companions.
- Going to the park
- Going for walks
- Will enjoy my company
As you can see, outlining is much easier when you have a visual aid to help you with the brainstorming process.
More Uses for Venn Diagrams
Besides its usefulness for planning essays, Venn Diagrams can be used for thinking through many other problems both at school and at home. For example:
- Planning a Budget: Create three circles for What I Want, What I Need, and What I Can Afford.
- Setting Priorities: Create circles for different types of priorities: School, Chores, Friends, TV, along with a circle for What I Have Time for This Week.
- Choosing Activities: Create circles for different types of activities: What I'm Committed to, What I'd Like to Try, and What I Have Time for Each Week.
- Comparing People's Qualities: Create circles for the different qualities you're comparing (ethical, friendly, good looking, wealthy, etc.), and then add names to each circle. Which overlap?
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How to Solve Venn Diagrams with 3 Circles
Venn diagrams with 3 circles: video lesson, what is the purpose of venn diagrams.
A Venn diagram is a type of graphical organizer which can be used to display similarities and differences between two or more sets. Circles are used to represent each set and any properties in common to both sets will be written in the overlap of the circles. Any property unique to a particular set is written in that circle alone.
For example, here is a Venn diagram comparing and contrasting dogs and cats.
The Venn diagram shows the following information:
- Have non-retractable claws
- Have round pupils
- Roam the street
- Have retractable claws
- Have slit pupils
Both dogs and cats:
- Can be pets
- Have 4 legs
A Venn diagram with three circles is called a triple Venn diagram.
A Venn diagram with three circles is used to compare and contract three categories. Each circle represents a different category with the overlapping regions used to represent properties that are shared between the three categories.
For example, a triple Venn diagram with 3 circles is used to compare dogs, cats and birds.
Dogs, cats and birds can all have claws and can also be pets.
Only birds:
- Have a beak
- Have 2 legs
Only both dogs and cats:
Only both dogs and birds:
Only both cats and birds:
- Don’t need walks
How to Make a Venn Diagram with 3 Circles
- Write the number of items belonging to all three sets in the central overlapping region.
- Write the remaining number of items belonging each pair of the sets in their overlapping regions.
- Write the remaining number of items belonging to each individual set in the non-overlapping region of each circle.
Make a Venn Diagram for the following situation:
30 students were asked which sports they play.
- 20 play basketball in total
- 16 play football in total
- 15 play tennis in total
- 10 play basketball and tennis
- 11 play basketball and football
- 9 play football and tennis
- 7 play all three
- Write the number of items belonging to all three sets in the central overlapping region
When making a Venn diagram, it is important to complete any overlapping regions first.
In this example, we start with the students that play all three sports. 7 students play all three sports.
The number 7 is placed in the overlap of all 3 circles. The shaded region shown is the overlapping area of all three circles.
2. Write the remaining number of items belonging each pair of the sets in their overlapping regions
There are 3 regions in which exactly two circles overlap.
There is the overlap of basketball and tennis, basketball and football and then tennis and football.
There are 10 students that play both basketball and tennis. The overlapping region of these two circles is shown below. We already have the 7 students that play all three sports in this region.
Therefore we only need 3 more students who play basketball and tennis but do not play football to make the total of this region add up to 10.
The next overlapping region of two circles is those that play basketball and football. There are 11 students in total that play both.
The overlapping region of the basketball and football circles is shown below.
There are already 7 students who play all three sports and so, a further 4 students must play both basketball and football but not tennis in order to make the total in this shaded region add up to 11 students.
The next overlapping region of two circles is those that play football and tennis. There are 9 students in total that play both.
The overlapping region of the football and tennis circles is shown below.
There are already 7 students who play all three sports and so, a further 2 students must play both football and tennis but not basketball in order to make the total in this shaded region add up to 9 students.
Write the remaining number of items belonging to each individual set in the non-overlapping region of each circle
There are three individual sets which are represented by the three circles. There are those that play basketball, football and tennis.
20 students play basketball in total. These 20 students are shown by the shaded circle below.
We already have 3, 7 and 4 students in the overlapping regions. This is a total of 14 students so far. We need a further 6 students who only play basketball in order for the numbers in this circle to make a total of 20.
The next individual sport is football. 16 students play football in total.
There are already 4, 7 and 2 students in the overlapping regions. This makes a total of 13 students so far.
3 more students are required to make the circle total up to 16. 3 students play only football and not basketball and tennis.
Finally, there are 15 students who play tennis shown by the shaded region below.
There are already 3, 7 and 2 students in the overlapping regions, making a total of 12 students.
A further 3 students are required to make the total of 15 students in this circle.
3 students play tennis but not basketball or football.
How to Solve a Venn Diagram with 3 Circles
To solve a Venn diagram with 3 circles, start by entering the number of items in common to all three sets of data. Then enter the remaining number of items in the overlapping region of each pair of sets. Enter the remaining number of items in each individual set. Finally, use any known totals to find missing numbers.
Venn diagrams are particularly useful for solving word problems in which a list of information is given about different categories. Numbers are placed in each region representing each statement.
100 people were asked which pets they have.
- 32 people in total just have a cat
- 18 people in total just have a rabbit
- 10 people have a dog and a rabbit
- 21 people have a dog and a cat
- 7 people have a cat and a rabbit
- 3 people own all three pets
How many people just have a dog?
Start by entering the number of items in common to all three sets of data
3 people own all three pets and so, a number 3 is written in the overlapping region of all three circles.
Then enter the remaining number of items in the overlapping region of each pair of sets
10 people have a dog and a rabbit.
Since 3 people are already in this region, 7 more people are needed.
21 people have a dog and a cat.
Since 3 people are already in this region, 18 more people are needed.
7 people have a cat and a rabbit.
Since 3 people are already in this region, 4 more people are needed.
Enter the remaining number of items in each individual set
32 people in total just have a cat.
There are already 18 + 3 + 4 = 25 people in this circle.
Therefore a further 7 people are needed in this circle to make 32.
7 people just own a cat and no other pet.
18 people in total just have a rabbit.
There are already 7 + 3 + 4 = 14 people in this circle.
Therefore a further 4 people are needed in this circle to make 18.
4 people just own a rabbit and no other pet.
Finally, use any known totals to find missing numbers
We are now told that 25 people own none of these pets. This means that a 25 is written outside of all of the circles but still within the Venn diagram.
The question requires the number of people who just own a dog.
There are 100 people in total and so, all of the numbers in the complete Venn diagram must add up to 100.
Adding the numbers so far, 3 + 7 + 4 + 18 + 4 + 7 + 25 = 68 people in total.
Since the numbers must add to 100, there must be a further 32 people who own a dog.
Now all of the numbers in the Venn diagram add to 100.
Venn Diagram with 3 Circles Template
Here is a downloadable template for a blank Venn Diagram with 3 circles.
How to Shade a Venn Diagram with 3 Circles
Here are some examples of shading Venn diagrams with 3 sets:
Shaded Region: A
Shaded Region: B
Shaded Region: C
Shaded Region: A∪B
Shaded Region: B∪C
Shaded Region: A∪C
Shaded Region: A∩B
Shaded Region: B∩C
Shaded Region: A∩C
Shaded Region: A∪B∪C
Shaded Region: A∩B∩C
Shaded Region: (A∩B)∪(A∩C)
VENN DIAGRAM WORD PROBLEMS WITH 3 CIRCLES
Let us consider the three sets A, B and C.
Set A contains a elements, B contains b elements and C contains c elements.
Both A and B contains w elements, B and C contains x elements, A and C contains y elements, all the three sets A, B and C contains z elements.
We can use Venn diagram with 3 circles to represent the above information as shown below.
Let us do the following changes in the Venn diagram.
We can get the following results from the Venn diagram shown above.
Number of elements related only to A is
= a - (w + y - z)
Number of elements related only to B is
= b - (w + x - z)
Number of elements related only to C is
= c - (y + x - z)
Number of elements related only to (A and B) is
= w - z
Number of elements related only to (B and C) is
= x - z
Number of elements related only to (A and C) is
= y - z
Number of elements related to all the three sets A, B and C is
Total number of elements related to all the three sets A, B and C is
= [a-(w+y-z)] + [b-(w+x-z)] + [c-(y+x-z)] + (w-z) + (x-z) + (y-z) + z
Example 1 :
In a survey of university students, 64 had taken mathematics course, 94 had taken chemistry course, 58 had taken physics course, 28 had taken mathematics and physics, 26 had taken mathematics and chemistry, 22 had taken chemistry and physics course, and 14 had taken all the three courses. Find how many had taken one course only.
Let M, C and P represent the courses Mathematics, Chemistry and Physics respectively.
Venn diagram related to the information given in the question:
From the venn diagram above, we have
No. of students who had taken only math = 24
No. of students who had taken only chemistry = 60
No. of students who had taken only physics = 22
Total no. of students who had taken only one course :
= 24 + 60 + 22
= 106
So, the total number of students who had taken only one course is 106.
Example 2 :
In a group of students, 65 play foot ball, 45 play hockey, 42 play cricket, 20 play foot ball and hockey, 25 play foot ball and cricket, 15 play hockey and cricket and 8 play all the three games. Find the total number of students in the group (Assume that each student in the group plays at least one game.)
Let F, H and C represent the games football, hockey and cricket respectively.
Venn diagram related to the information given in the question :
Total number of students in the group :
= 28 + 12 + 18 + 7 + 10 + 17 + 8
= 100
So, the total number of students in the group is 100.
Example 3 :
In a college, 60 students enrolled in chemistry,40 in physics, 30 in biology, 15 in chemistry and physics,10 in physics and biology, 5 in biology and chemistry. No one enrolled in all the three. Find how many are enrolled in at least one of the subjects.
Let C, P and B represent the subjects Chemistry, Physics and Biology respectively.
From the above Venn diagram, number of students enrolled in at least one of the subjects :
= 40 + 15 + 15 + 15 + 5 + 10 + 0
So, the number of students enrolled in at least one of the subjects is 100.
Example 4 :
In a town 85% of the people speak Tamil, 40% speak English and 20% speak Hindi. Also 32% speak Tamil and English, 13% speak Tamil and Hindi and 10% speak English and Hindi, find the percentage of people who can speak all the three languages.
Let T, E and H represent the people who speak the languages Tamil, English and Hindi respectively.
Let x be the percentage of people who speak all the three languages.
From the above Venn diagram, we can have
100 = 40 + x + 32 – x + x + 13 – x + 10 – x – 2 + x – 3 + x
100 = 40 + 32 + 13 + 10 – 2 – 3 + x
100 = 95 – 5 + x
100 = 90 + x
x = 100 - 90
x = 10%
So, the percentage of people who speak all the three languages is 10%.
Example 5 :
An advertising agency finds that, of its 170 clients, 115 use Television, 110 use Radio and 130 use Magazines. Also 85 use Television and Magazines, 75 use Television and Radio, 95 use Radio and Magazines, 70 use all the three. Draw Venn diagram to represent these data. Find
(i) how many use only Radio?
(ii) how many use only Television?
(iii) how many use Television and Magazine but not radio?
Let T, R and M represent the people who use Television, Radio and Magazines respectively.
From the above Venn diagram, we have
(i) Number of people who use only Radio is 10
(ii) Number of people who use only Television is 25
(iii) Number of people who use Television and Magazine but not radio is 15.
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- Three Circle Venn Diagrams
Three circle Venn Diagrams are a step up in complexity from two circle diagrams.
In this lesson we first look at how to read three circle diagrams. We then look at some word problems.
Reading Three Circle Diagrams
A Music Survey was carried out to find out what types of music a group of people liked.
The results were placed into the following three circle Venn Diagram.
To find out the total number of people surveyed, (or the E = Everything value), we add up all the numbers in the diagram. When we do this the answer is 70. This “E = Everything value” is also called “The Universal Set” of everything in the universe of our diagram. If we want to find the total number of people who like “Rock” music, then we add up all of the numbers in the “Rock” circle, including the areas where “Rock” overlaps with the other circles. The Total Rock People is: 16 + 2 + 8 + 5 = 31 people. We can also work out Probability or Odds from our Venn Diagram. For example we have found that 31 people out of 70 like Rock Music. So if we pick any one person at random from our group, the chances, or odds, or probability, that they will like Rock music is 31 out of 70, or 31 / 70, or 31/70 x100 = 44%. We can find the number of people who like all three types of music, by going to the centre of our diagram, where all three circles overlap. There are 8 people who like all three types of music.
Venn Diagram Word Problem One
This first problem is a fairly easy one, where all of the information we need has been given to us in the question. “A Class of 40 students completed a survey on what pets they like. The choices were: Cats, Dogs, and Birds. Everyone liked at least one pet. 10 students liked Cats and Birds but not dogs 6 students liked Cats and Dogs but not birds 2 students liked Dogs and Birds but not Cats 2 students liked all three pets 10 students liked Cats only 9 students liked Dogs only 1 student liked Birds only Represent these results using a three circle Venn Diagram.”
The type of three circle Venn Diagram we will need is the following:
This three circle word problem is an easy one. All of the number values for each section of the diagram have been given to us in the question. All we need to do is carefully put the number values onto the Diagram. We also need to check that all of the numbers add up to the total of 40 students when we are finished. The completed Venn Diagram is shown below.
Note that we do not need to color in and fill in the circles on Venn Diagrams. The following diagram is also correct and a fully acceptable answer.
Venn Diagram Word Problem Two This is a harder version of Problem One, where we are given less information in the question text. This means that we will need to do some working out steps to get to the final completed diagram.
We need the exact same type of Venn Diagram as for Question 1.
When we place what we know so far onto the diagram, this is what we have:
We now need to work through the other information in the word problem, one piece at a time. Usually in these problems we need to work on the overlapping parts in the centre of the diagram, and then work our way out to the “Cats Only”, “Dogs Only”, and “Birds Only” outer sections of the diagram. Remember: Work for the Inside Out. Here is what we will do next.
We can now fill in the answer of “2” onto the centre of our diagram.
We now have the Birds circle nearly completed. The only thing left to do is work out the “Birds Only” section, which we will now do.
We can now fill in the answer of “1” onto our diagram.
We now have the Birds circle completed. Next we work on the Cats Circle, following the exact same steps as we did on the Birds circle. First we need to work on the overlaps that involve Cats and other animals in the centre of the diagram.
This information refers to the following section of the Venn Diagram.
We can now fill in the answer of “6” onto our diagram.
We are nearly there ! We now only have “Cats Only” and “Dogs Only” to work out, and we have all the information we need to do this.
We can now place the “Cats Only” answer onto our diagram.
We now only need to work out “Dogs Only”.
Our Diagram for Problem Two is now finally complete.
Venn Diagram Word Problems Summary
Three Circle Videos
Here is a great Venn Diagrams video which also explains the “Inclusion / Exclusion” method.
This is an interesting three circles problem where they use a table of values to help with the working out.
Related Items
Introduction to Venn Diagrams Venn Diagram Word Problems Real World Venn Diagrams
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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
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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Well explained I hope more on this one
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Calculating a 3 Circle Venn Diagram only knowing A, B, and C?
I came across this question:
If there are 40 students in a class, 30 of them got A in Music, 33 of them got A in PE, and 37 of them got A in Art, at least how many students got all 3 As?
The first thing that came to my mind is to use Venn Diagram to solve it.
But most of the time when we solve Venn Diagram problems, $A \cap B, A \cap C, B \cap C$ , and $A \cap B \cap C$ are provided, namely, calculate the "total" amount of students in the class (40 in this case).
But this kind of problem must be able to be "reverse engineered" just like any other math problem.
So I went on and set
$A=30, B=33, C=37$
$A+B+C-A \cap B-A \cap C-B \cap C + A \cap B \cap C =40$ (Assuming that nobody got no As)
$\overline{A}=(B-A \cap B) + (C-A \cap C) - B \cap C=10$
$\overline{B}=(A-A \cap B) + (C-B \cap C) - A \cap C=7$
$\overline{C}=(B-B \cap C) + (A-A \cap C) - A \cap B=3$
And if I sum this up and add another $A+B+C$ , then it's $3(A+B+C-A \cap B-A \cap C-B \cap C)$ , then I could calculate $A \cap B \cap C$ from it.
But unfortunately, $3(A+B+C-A \cap B-A \cap C-B \cap C)$ turned out to be exactly 120, so $A+B+C-A \cap B-A \cap C-B \cap C=40$ and $A \cap B \cap C$ is $0$ !
So then I thought, maybe I shouldn't assume that nobody got no As.
And set a $\alpha$ as the number of students that got no As.
But soon find out this number will be "canceled" in the operation and therefore completely useless whatsoever!
Could somebody please be so kind and tell me where did I do wrong?
Much appreciated!
When I keep pondering through the problem, I first thought it might be the case just like the image below:
Where there "not necessary" have to have $A \cap B \cap C$ .
But when I double-check it by changing all 3 circles to 40 (meaning everybody got all straight As, thus $A \cap B \cap C$ should be 40 as well) and run through it with my calculation, I found out that $A \cap B \cap C$ is "still 0"!
Which means this way of calculation is "completely wrong"!
So I thought, maybe I could set it up like
and solving for g .
$a+b+f+g=30$
$b+c+d+g=33$
$d+e+f+g=37$
$a+b+c+d+e+f+g+h=40$
And found out that this linear equation simply "does not have enough information" to go on.
Could somebody please be so kind and teach me the correct way of doing it?
I found out why it's always 0:
Because those 3 equations I add up are originally $\overline{A}, \overline{B},$ and $\overline{C}$ , and if I add another $A+B+C$ to it, I'm essentially adding $A+\overline{A}+B+\overline{B}+C+\overline{C}$ , so of course it will be $3U (120)$ "no matter what"!
But, I still don't undertand why my "logic" is wrong, according to the equations, it "should" leave $A \cap B \cap C$ , but how come it somehow "disappeared" in the process? I really don't understand.
- combinatorics
- elementary-set-theory
- inclusion-exclusion
- 1 $\begingroup$ Notice that it asks ‘ at least how many got A’s in all 3 subjects?” $\endgroup$ – Bram28 Feb 21, 2022 at 13:59
- $\begingroup$ @Bram28 Please be so kind and take a look at my update. $\endgroup$ – Noob002 Feb 21, 2022 at 14:55
2 Answers 2
Earlier I provided an "answer" as a hint (in the form of a question). It was deleted as being more appropriate as a comment ("request for clarification") it was not a request for clarification. It was a hint in the form of a question.
"Hint: How many total A's were there out of how many total grades?"
The point is that there were $30+33+37=100$ total A's out of $3\cdot 40=120$ total grades. Hence there were $20$ grades that were not A's. If these grades were maximally spread around among the $40$ students, there would have been $20$ students who got a non A. (There may have been less such students, as some students could have gotten more than one non A.) But in any case that would mean that there are least $40-20=20$ students who got all A's.
- $\begingroup$ Wow! This is much easier to understand! I guess I went on the wrong direction "from the very beginning" because it was so similar to Venn Diagram problems and got mislead by it! Thank you very much for your help! $\endgroup$ – Noob002 Feb 21, 2022 at 19:31
- $\begingroup$ This is more like an "IQ test" than a regular question! ToT $\endgroup$ – Noob002 Feb 21, 2022 at 19:38
Let $x = |A\cap B\cap C|$ be the number of students that got all As.
We use the inclusion-exclusion formula,
$$ |A| + |B| + |C| - |A\cap B| - |B\cap C|-|C\cap A| + |A\cap B\cap C| = |A\cup B\cup C|,$$
$$x = |A\cup B\cup C| - |A| - |B| - |C| + |A\cap B| + |B\cap C|+|C\cap A|$$
To find a bound, we need to bound the intersections: using $$|A\cup B \cup C| \geq |A\cup B| = |A|+|B| - |A\cap B|,$$ we can rearrange to obtain $$|A\cap B| \geq |A| + |B| - |A\cup B \cup C|.$$ Plugging that (and analogous expressions) back into the equality for $x$ gives
$$x \geq |A\cup B \cup C| - |A|-|B|-|C|\\+ (|A|+|B|-|A\cup B \cup C| ) \\+ (|B|+|C|-|A\cup B \cup C| ) \\+ (|C|+|A|-|A\cup B \cup C| ) \\= |A|+|B|+|C| - 2|A\cup B\cup C|.$$
Thus $x \geq |A|+|B|+|C| - 2\cdot40 = 20$ , because $40 \geq |A\cup B\cup C|$ .
- $\begingroup$ Okay I might be wrong but I am fairly certain that while this bound is correct it is not tight. I think the correct lower bound is 20. You can see this by first noting that 30 people are in A. Then by assigning as many from B to not be in a we see that the intersection of A and B has at least 23, then applying the same logic with C we get 20. I cannot see any way to only have 17. $\endgroup$ – Fishbane Feb 21, 2022 at 17:12
- $\begingroup$ @Mentastin I think also that 20 is a tight bound. To show that 17 is a tight bound you should show us sets A,B,C where x=17. $\endgroup$ – miracle173 Feb 21, 2022 at 17:14
- $\begingroup$ Yes, that is correct. 20 is the tight bound (the discrepancy is due to first bounding by 37 and then by 40). I will edit. $\endgroup$ – Mentastin Feb 21, 2022 at 17:20
- $\begingroup$ @Mentastin Please rework your proof. But I can't see how you can claim that $|A\cup B\cup C| = 40$ $\endgroup$ – miracle173 Feb 21, 2022 at 17:35
- $\begingroup$ I don't, I claim that $|A\cup B\cup C| \leq 40$. So $x \geq |A|+|B|+|C|-2|A\cup B\cup C| \geq |A|+|B|+|C| -2*40 = 20$. $\endgroup$ – Mentastin Feb 21, 2022 at 17:39
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Sets Intersection: Intersection Of Three Sets
In these lessons, we will learn the intersection of three sets, how to shade regions of Venn Diagrams involving three sets and how to solve problems using the Venn Diagram of three sets (three circles).
Related Pages Intersection Of Two Sets Venn Diagrams More Lessons On Sets
Venn Diagrams Of Three Sets
The intersection of three sets X, Y and Z is the set of elements that are common to sets X, Y and Z. It is denoted by X ∩ Y ∩ Z.
Example: Draw a Venn diagram to represent the relationship between the sets X = {1, 2, 5, 6, 7, 9}, Y = {1, 3, 4, 5, 6, 8} and Z = {3, 5, 6, 7, 8, 10}
Solution: We find that X ∩ Y ∩ Z = {5, 6}, X ∩ Y = {1, 5, 6}, Y ∩ Z = {3, 5, 6, 8} and X ∩ Z = {5, 6, 7}
For the Venn diagram: Step 1: Draw three overlapping circles to represent the three sets.
Step 2: Write down the elements in the intersection X ∩ Y ∩ Z.
Step 3: Write down the remaining elements in the intersections: X ∩ Y, Y ∩ Z and X ∩ Z.
Step 4: Write down the remaining elements in the respective sets. Notice that you start filling the Venn diagram from the elements in the intersection first.
How To Shade Regions Of Venn Diagrams Involving Three Sets
Venn Diagrams: Shading Regions with Three Sets, Part 1 of 2 This video shows how to shade regions of Venn Diagrams involving three sets.
Example: Shade the indicated region:
- (A ∩ B) ∩ C
- (A ∪ B) ∩ C
Venn Diagrams: Shading Regions with Three Sets, Part 2 of 2 More example to show to shade regions of Venn Diagrams involving three sets.
Example: Shade the indicated region: 3) (A ∪ B)' ∩ C 4) (A' ∩ B') ∩ C'
How To Write An Expression For A Venn Diagram Region? Create an expression to represent the outlines part of the Venn Diagram shown.
How To Solve Word Problems With 3-Set Venn Diagrams?
Venn Diagram Problem With 3 Circles Use the given information to fill in the number of elements in each region of the Venn Diagram. This video solves two problems using Venn Diagrams. One with two sets and one with three sets.
Example 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
- How many signed up only for a math class?
- How many signed up only for an English class?
- How many signed up for math or English?
- How many signed up for neither math nor English?
Example 2: 100 were students 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 How many students took none of the three subjects? How many students took PE, but not Bio or Eng? How many students took Gio and PE but not Eng?
How To Solve A Venn Diagram Problem Involving Three Sets? Example: 110 college freshmen were surveyed 25 took physics 45 took biology 45 took mathematics 10 took physics and mathematics 8 took biology and mathematics 6 took physics and biology 5 took all three a. How many students took biology, but neither physics nor mathematics? b. How many students took biology, physics or mathematics? c. How many students did not take any of the three subjects?
How to fill up a 3-circle Venn Diagram? In this video we go over a basic word problem involving three sets. We use a Venn diagram to answer the series of questions.
Venn Diagram Shading Calculator Or Solver Enter an expression like (A Union B) Intersect (Complement C) to describe a combination of two or three sets and get the notation and Venn diagram. Use parentheses, Union, Intersection, and Complement.
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3 Circle Venn Diagram Worksheets
Welcome to our 3 Circle Venn Diagram Worksheet collection for 3rd and 4th graders. Here you will find a wide range of free venn diagram sheets which will help your child learn to classify a range of objects, shapes and numbers using venn diagrams.
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3 Circle Venn Diagram
What is a 3 circle venn diagram.
A 3 Circle Venn diagram is a way of classifying groups of objects with the same properties.
It has three circles that intersect each other.
Each circle has its own set of properties of things that go into the circle, e.g. odd numbers or shapes with right angles.
There is also a space outside the circles where objects that do not fit any of the properties can go.
The diagram below shows you how a venn diagram with 3 circles works.
Venn Diagram Worksheets
Need to practice using venn diagrams?
Looking for some worksheets to help you to classify a range of different objects using venn diagrams?
Then this page should hopefully be what you are looking for!
The worksheets on this page have been split into 2 sections:
- 3rd grade venn diagrams
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Each sheet consists of sorting either a table of facts, shapes or numbers.
Each sheet contains one set of objects and a 3 circle venn diagram to sort the objects with.
There is a progression from easier to harder sheets within each section.
Using the sheets in this section will help your child to:
- practice using three circle venn diagrams;
- converting data from a table to a venn diagram;
- practice classifying a range of objects using different properties;
- classifying numbers using properties such as prime, multiples and factors.
Help using 3 Circle Venn Diagrams
Looking for some help using 3 circle venn diagrams?
This short video will hopefully show you all you need to master them!
3rd Grade Venn Digrams
- Venn Diagram 3 Circles Sheet 3:1
- PDF version
- Venn Diagram 3 Circles Sheet 3:2
- Venn Diagram 3 Circles Sheet 3:3
- Venn Diagram 3 Circles Sheet 3:4
4th Grade Venn Digrams
- Venn Diagram 3 Circles Sheet 4:1
- Venn Diagram 3 Circles Sheet 4:2
- Venn Diagram 3 Circles Sheet 4:3
- Venn Diagram 3 Circles Sheet 4:4
Looking for help with venn diagrams?
If venn diagrams are a bit of a mystery to you and you would like some more support, then try our page all about venn diagrams.
There is a video to watch with some simple examples worked through.
- What is a venn diagram page
Looking for some 2 Circle Venn Diagram Worksheets?
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- Venn Diagram Worksheets 3rd Grade
- Venn Diagram Worksheet 4th Grade
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- 3rd Grade Rounding Inequalities Multiples Worksheets
- Factors and Multiples Worksheet
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3-circle Venn diagram
A 3-circle Venn diagram, named after the English logician Robert Venn, is a diagram that shows how the elements of three sets are related using three overlapping circles.
When the three circles in a Venn diagram overlap, the overlapping parts contain elements that are common to any two circles or all the three circles.
Some important observations
- Elements in A and B = elements in A and B only plus elements in A, B, and C.
- Elements in B and C = elements in B and C only plus elements in A, B, and C.
- Elements in A and C = elements in A and C only plus elements in A, B, and C.
Example showing how to create a 3-circle Venn diagram
We can create a 3-circle Venn diagram to show the relationships between the factors of 30, 40, and 100
Factors of 30 : 1, 2, 3, 5, 6, 10, 15, 30
Factors of 40 : 1, 2, 4, 5, 8, 10, 20, 40
Factors of 100 : 1, 2, 4, 5, 10, 20, 25, 50, 100
Now take this quiz about 3-circle venn diagram
Intersection of sets
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In this video we go over a basic word problem involving three sets. We use a venn diagram to answer the series of questions.
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This three circle word problem is an easy one. All of the number values for each section of the diagram have been given to us in the question.
Venn Diagram Examples, Problems and Solutions · n(C) = percentage of women who like watching comedy = 52% · n(F) = percentage of women who like watching fantasy =
But most of the time when we solve Venn Diagram problems, A∩B,A∩C,B∩C, and A∩B∩C are provided, namely, calculate the "total" amount of
The intersection of three sets X, Y and Z is the set of elements that are common to sets X, Y and Z. It is denoted by X ∩ Y ∩ Z.
For 3 variables, first draw 3 circles like this: Now fill in the data given. The total region covered by the 3 circles is 81%, so outside region is 100-81
A 3 Circle Venn diagram is a way of classifying groups of objects with the same properties. It has three circles that intersect each other. Each circle has its
A 3-circle Venn diagram, named after the English logician Robert Venn, is a diagram that shows how the elements of three sets are related using three