Venn Diagram Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Venn Diagram.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A diagram using overlapping circles to visually represent sets and their relationships such as union, intersection, and complement.

Each circle represents a set; overlapping regions show shared elements; the rectangle border is the universal set.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A Venn diagram draws sets as circles so union, overlap, and outside become visible regions.

Common stuck point: The procedure for venn diagram is the easy part; the trap is writing each circle's full total in its 'only' region. Asking "Are there overlapping categories whose regions I need to picture and count separately?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are there overlapping categories whose regions I need to picture and count separately?

Worked Examples

Example 1

easy

In a class of 30 students, 18 study French, 12 study Spanish, and 5 study both. Use a Venn diagram to find how many study neither language.

Answer

5 \text{ students study neither language}

First step

Let

F

= French students,

S

= Spanish students. Place

|F \cap S| = 5

in the overlap region.

Full solution

2
French only: $18 - 5 = 13$ . Spanish only: $12 - 5 = 7$ . So $|F \cup S| = 13 + 5 + 7 = 25$ .
3
Students studying neither: $30 - 25 = 5$ .

A Venn diagram partitions the universal set into non-overlapping regions. The inclusion-exclusion principle gives

|F \cup S| = |F| + |S| - |F \cap S|

Example 2

medium

Given

|A| = 20

|B| = 15

|A \cup B| = 28

, find

|A \cap B|

using the inclusion-exclusion principle.

Example 3

medium

In a town survey of 200 households, 120 own a dog, 90 own a cat, and 60 own both. How many own at least one of the two?

Example 4

medium

Use a Venn diagram to verify:

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

(distributive law).

Example 5

hard

Use a Venn diagram to verify De Morgan's law:

(A\cup B)^c = A^c \cap B^c

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

In a survey of 50 people, 30 like tea, 25 like coffee, and 10 like both. How many like only tea? Only coffee? Neither?

Example 2

medium

In a class of 30 students, 18 study French, 12 study Spanish, and 5 study both. How many study neither language?

Example 3

easy

In a two-circle Venn diagram for sets

A

and

B

, which region represents

A \cap B

Example 4

easy

In a Venn diagram, which region represents

A \cup B

Example 5

easy

In a Venn diagram with universe

U

drawn as a rectangle, what does the region OUTSIDE both circles represent?

Example 6

easy

In a Venn diagram, which region is 'in

A

but not in

B

Example 7

easy

If circles

A

and

B

do NOT overlap in a Venn diagram, what is

A \cap B

Example 8

easy

If circle

A

is drawn entirely inside circle

B

in a Venn diagram, what relationship holds?

Example 9

easy

A Venn diagram shows 5 in 'A only', 3 in the overlap, 4 in 'B only'. How many elements are in

A

Example 10

easy

A Venn diagram shows 5 in 'A only', 3 in the overlap, 4 in 'B only'. How many are in

A \cup B

Example 11

medium

In a class of 30, a Venn diagram shows 14 take art, 18 take music, 6 take both. How many take art only?

Example 12

medium

In the same class of 30 (14 art, 18 music, 6 both), how many take neither?

Example 13

medium

A Venn diagram for

A

and

B

has 'A only'

=x

, overlap

=7

, 'B only'

=9

, and

|A|=15

. Find

x

Example 14

medium

Three-circle Venn diagram: the very center region represents which set operation on

A

B

C

Example 15

medium

In a Venn diagram, which region represents elements in exactly one of

A

B

(the symmetric difference)?

Example 16

medium

A Venn diagram shows: only

A=10

, only

B=8

, both

=5

, neither

=7

. What is the total number of elements in the universe?

Example 17

medium

Two circles overlap in a Venn diagram. Does the overlap GUARANTEE the sets share elements?

Example 18

challenge

In a class of 40, a three-circle Venn diagram has: all three

=2

, exactly two subjects

=9

total, exactly one subject

=20

total. How many take none?

Example 19

challenge

A three-circle Venn diagram has

|A|=|B|=|C|=20

, each pairwise overlap (total)

=8

, and all three

=3

. Find the number in exactly one circle.

Example 20

challenge

Use a Venn diagram argument to explain why

|A \cup B| = |A| + |B| - |A \cap B|

Example 21

medium

A Venn diagram shows 'A only'

=6

, overlap

=4

, 'B only'

=5

. Find

|A \cap B|

Example 22

medium

A Venn diagram shows 'A only'

=6

, overlap

=4

, 'B only'

=5

. Find

|B|

Example 23

easy

A Venn diagram has 'A only'

= 7

, overlap

= 4

, 'B only'

= 6

. Find

|A|

Example 24

easy

A Venn diagram has 'A only'

= 7

, overlap

= 4

, 'B only'

= 6

, and

5

outside both. Find the size of the universe

U

Example 25

easy

Use inclusion–exclusion:

|A|=12,\ |B|=9,\ |A\cap B|=4

. Find

|A\cup B|

Example 26

easy

In a class of 25,

13

play soccer,

11

play basketball,

4

play both. How many play only soccer?

Example 27

easy

Same class (25 students, 13 soccer, 11 basketball, 4 both). How many play neither sport?

Example 28

medium

In a survey of 80, 45 like pizza, 50 like burgers, 28 like both. How many like exactly one of the two?

Example 29

medium

|A| = 18

|B| = 12

, and

A \cap B = \emptyset

. Find

|A \cup B|

Example 30

medium

|A \cup B| = 30

|A| = 22

|B| = 17

. Find

|A \cap B|

Example 31

medium

In a 3-circle Venn diagram, which region represents 'in

A

and

B

but not

C

Example 32

medium

Survey of 60 students: 30 take French (F), 25 Spanish (S), 20 German (G); pairwise overlaps F∩S = 10, F∩G = 8, S∩G = 6; all three = 4. How many take at least one language?

Example 33

medium

Continuing the survey of 60 (F=30, S=25, G=20; pairwise 10, 8, 6; all three 4). How many take NONE of the three languages?

Example 34

medium

A Venn diagram has 'A only' =

2x

, overlap =

x

, 'B only' =

3x

. If

|A \cup B| = 36

, find

x

Example 35

medium

From a class of 50: 28 like math, 22 like science, 15 like both. How many like math but not science?

Example 36

hard

In a class of 40: 22 play piano, 18 play guitar, 14 play drums, 8 play piano and guitar, 6 play piano and drums, 5 play guitar and drums, 3 play all three. How many play none?

Example 37

hard

Same class (40 students,

|P\cup G\cup D|=38

). How many play EXACTLY one instrument?

Example 38

hard

In a survey of 100 readers: 60 read fiction, 50 read non-fiction. The number who read both is unknown. What are the smallest and largest possible values of 'both'?

Example 39

hard

|A|=20,\ |B|=15,\ |C|=10,\ |A\cap B|=8,\ |A\cap C|=5,\ |B\cap C|=4,\ |A\cap B\cap C|=2

. Find

|A\cup B\cup C|

Example 40

challenge

Three sets

A,B,C

satisfy

|A\cap B|+|A\cap C|+|B\cap C|=42

and

|A\cap B\cap C|=10

. How many elements lie in exactly two of the three sets?

Example 41

challenge

Why can't a standard Venn diagram with circles correctly represent all

2^n

regions for

n \ge 4

sets?

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Related Concepts

Set Union Intersection

Background Knowledge

These ideas may be useful before you work through the harder examples.

setunionintersection