Venn Diagram Examples: 2 and 3 Sets, With Solutions

The best way to learn a Venn diagram is to work through Venn diagram examples, starting with the easiest. This guide does that. Every example has a question, a solution and a diagram, and each new one adds one idea: two sets of numbers, an everyday pair, a survey with counts, a third circle, an empty overlap, and ideas for kids. You will also get the set symbols in plain English and a few practice questions. These are Venn diagram examples with solutions you can check yourself, because every item is sorted by a rule.

What a Venn Diagram Shows

According to Wikipedia, a Venn diagram, also called a set diagram or logic diagram, shows all possible logical relations between a finite collection of sets. Each set is drawn as a closed curve, usually a circle. A point inside a circle stands for an element of that set, and a point outside stands for something that is not in it.

Sets, Circles and the Overlap

A set is a group of things. The things in it are its elements. Each circle in the diagram holds one set. Where two circles overlap, you find the elements that belong to both sets.

The Four Regions of Two Circles

The open textbook Contemporary Mathematics from OpenStax says a Venn diagram with two overlapping sets breaks the universal set into four regions. Three of them are inside the circles: the part of A only, the part of B only, and the overlap. The fourth is the space outside both circles, which holds everything that is in neither set.

Keep that outside region in mind. It matters in the examples below, and our own tool handles it in a particular way that we explain in the survey example.

Example 1: Two Sets Defined by a Rule

The easiest sets to check are numbers, because a rule tells you exactly who is in. Here is the first question.

Question. Look at the numbers 1 to 12. Make a two circle Venn diagram of the multiples of 2 and the multiples of 3. Which numbers are in both sets? Which are in neither?

The Solution

1. List each set. The multiples of 2 up to 12 are 2, 4, 6, 8, 10 and 12. The multiples of 3 are 3, 6, 9 and 12.
2. Find the numbers on both lists. Only 6 and 12 appear twice. They go in the overlap.
3. Fill in the rest. The numbers 2, 4, 8 and 10 are multiples of 2 only. The numbers 3 and 9 are multiples of 3 only.
4. Find who is left. Of the numbers 1 to 12, we have placed 2, 3, 4, 6, 8, 9, 10 and 12, which is 8 numbers. The other 4 numbers, 1, 5, 7 and 11, are in neither set.

Notice that a number such as 6 is written once, even though it is on two lists. An element is in the diagram once, and the overlap is where its two memberships meet.

Example 2: An Everyday Two-Set Diagram

Sets do not have to be numbers. Anything you can list works, as long as you can tell whether an item is on a list.

Question. One pizza has cheese, tomato, mushroom, olives and basil. A second pizza has cheese, pepperoni, mushroom and onion. Which toppings are on both pizzas? Which are only on Pizza 2?

Solution. Read each topping on Pizza 2 and ask whether it is also on Pizza 1. Cheese is, and mushroom is. So the overlap holds cheese and mushroom. Pepperoni and onion are not on Pizza 1, so they are on Pizza 2 only. The tomato, olives and basil are on Pizza 1 only.

Count the different toppings in all: 3 on Pizza 1 only, 2 on both, and 2 on Pizza 2 only, which makes 7. Pizza 1 has 5 toppings and Pizza 2 has 4, which add to 9, so two toppings were counted twice. Hold on to that idea. It is the next example.

Example 3: A Survey With Counts

Real problems often give you numbers, not names. This one is the classic exam question, and it is where a Venn diagram earns its keep.

Question. A class of 30 students is asked about sports. 18 like soccer. 14 like basketball. 7 like both. How many like only soccer? Only basketball? At least one of the two? Neither?

The Solution, Step by Step

1. Start with the overlap. 7 students like both, so write 7 in the middle.
2. Subtract it from each circle. The soccer total of 18 includes those 7, so 18 minus 7 gives 11 who like only soccer. The basketball total of 14 also includes them, so 14 minus 7 gives 7 who like only basketball.
3. Add the three parts inside the circles. 11 plus 7 plus 7 is 25 students who like at least one sport.
4. Find the outside. The class has 30 students, so 30 minus 25 leaves 5 who like neither.
5. Check. 11 plus 7 plus 7 plus 5 is 30, the whole class.

The most common slip is to write 18 in the soccer part and 14 in the basketball part and forget that both totals contain the overlap. Always subtract the overlap first.

The Two-Set Addition Rule

Step 3 has a shortcut. You do not need the regions to count everyone inside the circles. OpenStax gives the rule: the number of elements in the union of two sets equals the number in A, plus the number in B, minus the number in both. Adding A and B counts the overlap twice, so you subtract it once.

For the survey, 18 plus 14 minus 7 is 25. It matches the 25 we found region by region. For the numbers example, 6 plus 4 minus 2 is 8. For the pizzas, 5 plus 4 minus 2 is 7. If two sets share nothing, the overlap is 0 and the sizes just add, which is the fifth example below.

How the Tool Shows Counts and the Outside

Our tool sorts items into regions, so to get counts we typed one placeholder ID per student: S1 to S18 in the soccer column and S12 to S25 in the basketball column. The seven IDs that appear in both columns land in the overlap, and switching the display to counts turns each region into a number.

The tool has no rectangle and no outside region. An item that is in no column is simply not in the diagram, so the 5 students who like neither are not counted anywhere on the picture. Write that number in your subtitle or beside the chart, as we did here, or the reader will think the class had 25 students.

Example 4: A Three-Set Example

With three circles you have seven regions inside plus the outside. The method is the same: test each item against each rule. Here is one worked example. Our guide to the 3 circle Venn diagram names every region and covers the counting formula in full.

Question. Sort the numbers 6 to 15 into three sets: multiples of 3, multiples of 5, and odd numbers. Where does each number go?

Solution. Test each number against the three rules.

• Multiples of 3 only: 6 and 12. They are multiples of 3 and even.
• Multiples of 5 only: 10. It is a multiple of 5 and even.
• Odd only: 7, 11 and 13.
• Multiples of 3 and odd, not of 5: 9.
• All three: 15, which is a multiple of 3, a multiple of 5 and odd.
• Multiples of 5 and odd, not of 3: none. The only odd multiple of 5 in this range is 15, and it is also a multiple of 3.
• Multiples of 3 and of 5 but not odd: none, for the same reason. Two of the seven regions are empty, and that is a correct answer, not a gap in the work.
• Outside: 8 and 14.

Check the total: 2 plus 1 plus 3 plus 1 plus 1 plus 0 plus 0 is 8 numbers inside the circles, and 2 outside makes 10, which is how many numbers there are from 6 to 15.

One reading habit to learn now: the phrase “odd numbers that are multiples of 3” means 9 and 15, and it includes the center. The phrase “only” removes it. That is why 15 is not in the “3 and odd” region above.

Example 5: When the Overlap Is Empty

An empty overlap looks like a mistake, but it is often the answer. OpenStax calls two sets with nothing in common disjoint, and says their intersection is the empty set.

Question. Look at the numbers 1 to 30. Which numbers are multiples of both 5 and 7? How many numbers are in the union?

Solution. The multiples of 5 up to 30 are 5, 10, 15, 20, 25 and 30, which is 6 numbers. The multiples of 7 are 7, 14, 21 and 28, which is 4. Compare the lists: no number appears twice. A number that is a multiple of both 5 and 7 has to be a multiple of 35, and 35 is bigger than 30. So the overlap is empty, and the union is 6 plus 4 minus 0, which is 10 numbers.

Our tool always draws two overlapping circles, even for disjoint sets, so an empty overlap is drawn as a blank region, or as a 0 in counts mode. Some textbooks draw disjoint sets as two separate circles instead. Both pictures say the same thing.

Venn Diagram Examples for Kids

Kids do best when each item can be sorted by looking at it and by one clear rule. These two examples fit that, and they need no arithmetic.

Letters in Two Words

Question. Which letters do the words APPLE and PEAR share? Which letters are only in APPLE?

Solution. List each different letter once. APPLE has A, P, L and E, since the second P is not a new letter. PEAR has P, E, A and R. The letters on both lists are A, P and E. L is only in APPLE, and R is only in PEAR. The two words have 4 different letters each, and 3 are shared, so together they use 4 plus 4 minus 3, or 5 different letters.

Shapes and Colors

Here is an idea you can do on paper with cut-out shapes. Lay out five shapes: a red circle, a red square, a blue circle, a blue triangle and a yellow square. Make one set of red shapes and one set of round shapes. The red circle is red and round, so it goes in the overlap. The red square goes in red only, and the blue circle in round only. The blue triangle and the yellow square are neither, so they go outside the circles. Ask a child why the blue triangle has no place inside.

Here are two more sets a child can check by looking: animals that fly and animals that swim, or foods that are fruit and foods that are red. Choose items that clearly fit one side or both, and avoid items where adults would argue. Those tips are our own advice.

Set Symbols in Plain English

Textbook questions use a short set of symbols. Venn diagram symbols are not a separate language: they are the ordinary set notation, and a Venn diagram is a picture of them. Here they are, with the multiples example, where the universal set is the numbers 1 to 12, A is the multiples of 2 and B is the multiples of 3.

Union and Intersection

OpenStax defines the union of two sets as everything in the first set, the second set, or both. It works like a logical inclusive OR. The intersection is what the two sets have in common, so a member must be in both. It works like AND. In a two circle diagram, the intersection is the overlap, and the union is both circles together.

Complement and Universal Set

OpenStax describes the universal set as the largest set under consideration, drawn as a rectangle with the other sets as circles inside it. The complement of a set A is every member of the universal set that is not in A. A set and its complement share nothing. In our example, the complement of the multiples of 2 is the odd numbers from 1 to 11.

Notation differs from book to book. OpenStax writes the complement with a prime mark, A′, and notes that a set-practice app it recommends writes it with a small letter C after the set. Check which one your class uses.

The Empty Set and Disjoint Sets

The empty set, ∅, has no elements. Two sets are disjoint when they have none in common, so their intersection is ∅. That is exactly what happened in Example 5. Our tool draws no rectangle for the universal set, so if your teacher expects one, sketch it around the picture and write the outside count in it.

Venn Diagram or Euler Diagram?

The two are close cousins. According to Wikipedia, a Venn diagram overlaps its curves in every possible way, showing every relation between the sets. That makes it a special case of an Euler diagram, which does not have to show every relation. So if two sets could never share anything, an Euler diagram would draw them apart, while a Venn diagram keeps the overlap and leaves it empty. Example 5 is the second case.

A Short History and Where Venn Diagrams Are Used

Where the Idea Came From

Wikipedia says John Venn popularized the diagrams in his book Symbolic Logic, published in 1881. Similar ideas came earlier from Christian Weise in 1712 and Leonhard Euler in 1768.

Where You Meet Them Today

They are used to teach elementary set theory and to show simple set relationships in probability, logic, statistics, linguistics and computer science.

Practice Questions

Answer these from the diagrams above, then check with the solutions.

1. In Example 1, how many numbers are multiples of 2 but not of 3? 4 numbers: 2, 4, 8 and 10.
2. In Example 1, how many numbers from 1 to 12 are in neither set? 4 numbers: 1, 5, 7 and 11.
3. In Example 2, how many different toppings are there on the two pizzas together? 7. Use 5 plus 4 minus 2.
4. In Example 3, how many students like only basketball? 7, which is 14 minus the 7 who like both.
5. In Example 3, how many students like at least one sport? 25. Check it with 18 plus 14 minus 7.
6. In Example 4, which numbers are multiples of 3 or multiples of 5? 6, 9, 10, 12 and 15, which is 5 numbers. Check: 4 plus 2 minus 1 is 5.
7. In Example 5, what is the size of the union? 10, because the overlap is 0.
8. In the kids example, how many different letters do APPLE and PEAR use? 5: A, P, L, E and R.

Start From a Blank Two Circle Diagram

The Template

Open the template to try your own sets. It has two circles named Set A and Set B and four placeholder items, one in each region and one extra in the overlap. Rename the sets, type your own items, and the diagram updates as you type.

Open the blank two circle template

What the Tool Does and Does Not Do

Here is what the tool does, so you know what to expect:

• Draws two or three circles. Choose them under Number of circles.
• Sorts your items into regions for you. Type an item in every column where it belongs.
• Shows item names or counts, using the Inside the circles setting, and can add the total next to each set name.
• Draws all circles the same size. The diagram is not drawn to scale, so the area of a region does not show how many items it holds. Wikipedia has a separate name for the kind whose areas do match the counts: an area-proportional, or scaled, Venn diagram. Ours is not one.
• Matches items without regard to upper and lower case, and counts an item once per set.
• Does not draw an outside region or count the items that are in neither set.

The Venn diagram maker is free and runs in your browser. For three sets and the formula that goes with them, see the guide to the 3 circle Venn diagram.

Common Mistakes to Avoid

• Forgetting the outside. The items in neither set are part of the total. Write that number down yourself.
• Adding two totals without subtracting the overlap. Both totals include the shared items, so subtract them once.
• Treating the overlap of two circles as empty. A Venn diagram keeps the overlap even when nothing is in it. Say so with a 0.
• Guessing membership. For each item, ask a yes or no question about every set, one at a time.

Venn diagrams are one of several ways to compare groups. Our overview of types of graphs and charts shows where they sit next to bar charts, pie charts and timelines. When all you need is to compare how big each group is, a bar graph is simpler: see our bar graph examples.

Summary

Start each Venn diagram example the same way: list the sets, find the items on both lists, and fill the overlap first. For counts, subtract the overlap from each total, and use the two-set rule to check that the parts add up. Remember the outside, because your diagram might not show it. Then try your own sets in the Venn diagram maker.