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Math Coach Amy
Amy Ferguson Moncure

Geometry · Unit 1

Venn Diagrams

Count regions in two- and three-set diagrams, and read union, intersection, and complement.

Jump to section:

Key Ideas Easy Medium Hard

A ∩ B intersection (overlap)

A ∪ B union (married)

A′ not in A

What to Remember

The universe is the box

The rectangle is the universe ξ. Everyone belongs inside the box. The number in the lower-right corner is outside both circles, but still in the universe. Each region is counted once.

Just / only

“Just cats” or “only cats” is the Cats region outside Dogs. In the example, that is 8.

AND ∩ and overlap (intersection)

A ∩ B is the intersection (overlap). In the example, cats and dogs is 3.

OR ∪ (think union or married)

A ∪ B is the union (married): everyone in A, in B, or in both. In the example, cats or dogs is 8 + 3 + 7 = 18.

Level 1

Easy

Two circles. Just/only, intersection (overlap), and union (married).

Example
The diagram shows students in Yoga and Pilates.
How many students are in just Yoga?

Just Yoga is the Yoga-only region.

22

Answer: 22

1
Use the Yoga and Pilates diagram.
How many students take Yoga and Pilates (the intersection)?
Show Answer

Yoga ∩ Pilates is the intersection (overlap).

9

Answer: 9

2
Use the Yoga and Pilates diagram.
How many students take Yoga or Pilates (the union / married)?
Show Answer

Yoga ∪ Pilates is the union (married): everyone in Yoga, in Pilates, or in both.

22 + 9 + 15 = 46

Answer: 46

3
Use the Yoga and Pilates diagram.
How many students take neither class?
Show Answer

Neither is the outside region, still inside the universe box.

8

Answer: 8

4
The diagram shows visitors in the Library and the Cafe.
How many visitors are in just the Library?
Show Answer

Just Library is the Library-only region.

31

Answer: 31

5
Use the Library and Cafe diagram.
How many visitors are in the Cafe?
Show Answer

Cafe is the whole Cafe circle: just Cafe plus the intersection (overlap).

19 + 12 = 31

Answer: 31

6
Here is a Venn diagram.
Write down the numbers that are in set D.
Show Answer

D is the whole D circle: the intersection (overlap) and just D.

2, 10, 4, 8

Answer: 2, 4, 8, 10

7
Use the C and D diagram.
Write down the numbers that are in C ∪ D (union / married).
Show Answer

C ∪ D is the union (married): every number in C, in D, or in both.

3, 7, 2, 10, 4, 8

Answer: 2, 3, 4, 7, 8, 10

8
Use the C and D diagram.
Write down the numbers that are in C′.
Show Answer

C′ is everyone not in C: just D, plus the outside of both circles.

4, 8, 1, 9

Answer: 1, 4, 8, 9

Level 2

Medium

Place numbers, fill a diagram from a story, and read a union or complement.

Example
The diagram shows club membership.
How many students are in just Art?

Just Art is the Art-only region.

20

Answer: 20

1
ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. Even = even numbers. Mult. of 3 = multiples of 3. Complete the Venn diagram.
Show Answer

Even, not a multiple of 3: 2, 4, 8, 10.

Both (intersection): 6, 12.

Multiple of 3, not even: 3, 9.

Outside both circles: 1, 5, 7, 11.

Answer: 2, 4, 8, 10 in just Even; 6, 12 in the intersection; 3, 9 in just Mult. of 3; 1, 5, 7, 11 outside

2
There are 70 students in the grade. 8 students study chorus and band. 24 students only study chorus. 6 students do not study chorus or band. Complete the Venn diagram.
How many students study only band?
Show Answer

Intersection (overlap): 8.

Just chorus: 24.

Outside: 6.

Only band = 70 − 8 − 24 − 6 = 32.

Answer: 32

3
Here is a Venn diagram.
Write down the numbers that are in A ∩ B (intersection).
Show Answer

A ∩ B is the intersection (overlap).

4, 9

Answer: 4, 9

4
Use the A and B diagram.
Write down the numbers that are in A ∪ B (union / married).
Show Answer

A ∪ B is the union (married): every number in A, in B, or in both.

16, 25, 36, 4, 9, 5

Answer: 4, 5, 9, 16, 25, 36

5
Use the A and B diagram.
Write down the numbers that are in A′.
Show Answer

A′ is everyone not in A.

5, 17, 40

Answer: 5, 17, 40

6
A rec center ran pottery and painting. 80 people visited. 22 people took pottery. 9 people took both classes. 41 people took neither class. Draw a Venn diagram for this information.
Show Answer

Intersection (overlap): 9.

Just pottery: 22 − 9 = 13.

Neither: 41.

Just painting: 80 − 13 − 9 − 41 = 17.

Answer: just pottery 13, both 9, just painting 17, neither 41

7
Use the Art, Band, and Choir diagram.
How many students are in Band or Choir (union / married)?
Show Answer

Band ∪ Choir is the union (married): everyone in Band, in Choir, or in both.

15 + 9 + 5 + 6 + 11 + 7 = 53

Answer: 53

8
Use the Art, Band, and Choir diagram.
How many students are not in Band?
Show Answer

Band′ is everyone outside the Band circle.

Band has 15 + 9 + 5 + 6 = 35.

The universe is 20 + 15 + 11 + 9 + 7 + 6 + 5 + 10 = 83.

83 − 35 = 48

Answer: 48

Level 3

Hard

Three circles, complements, and listing the numbers in a region.

Example
The diagram shows sports sign-ups.
Find Soccer ∩ Tennis (intersection).

Soccer ∩ Tennis is the intersection (overlap), including the center.

8 + 3 = 11

Answer: 11

1
Use the Soccer, Tennis, and Volleyball diagram.
How many students have just soccer?
Show Answer

Just soccer is the Soccer-only region.

16

Answer: 16

2
Use the sports diagram.
Find Soccer ∩ Volleyball′.
Show Answer

Soccer ∩ Volleyball′ is Soccer but not Volleyball: just Soccer, plus Soccer ∩ Tennis.

16 + 8 = 24

Answer: 24

3
Use the sports diagram.
Find (Soccer ∪ Volleyball) ∩ Tennis.
Show Answer

First form the union (married) of Soccer and Volleyball.

Then keep the intersection with Tennis.

8 + 3 + 4 = 15

Answer: 15

4
Use the sports diagram.
Find (Tennis ∪ Volleyball)′ ∩ Soccer.
Show Answer

(Tennis ∪ Volleyball)′ is outside both Tennis and Volleyball.

Intersect that with Soccer: just Soccer.

16

Answer: 16

5
Here is a Venn diagram.
Write down the numbers that are in A ∩ B′.
Show Answer

A ∩ B′ is just A.

6, 13

Answer: 6, 13

6
Use the same diagram.
Write down the numbers that are in A′ ∪ B′.
Show Answer

A′ ∪ B′ is the union (married) of the two complements: everyone except the intersection.

6, 13, 2, 8, 14, 1, 15

Answer: 1, 2, 6, 8, 13, 14, 15

7
ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}. A = multiples of 4. B = multiples of 6. Complete the Venn diagram.
Show Answer

Just A: 4, 8.

Intersection (overlap): 12.

Just B: 6.

Outside: 1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 15.

Answer: just A: 4, 8; intersection: 12; just B: 6; outside: 1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 15

8
A fair had a ferris wheel and a carousel. 120 people came. 35 rode the ferris wheel. 14 rode both. 50 rode neither. Draw a Venn diagram for this information.
Show Answer

Intersection (overlap): 14.

Just ferris wheel: 35 − 14 = 21.

Neither: 50.

Just carousel: 120 − 21 − 14 − 50 = 35.

Answer: just ferris 21, both 14, just carousel 35, neither 50

9
Here is a Venn diagram.
Write down the numbers that are in P ∩ Q′.
Show Answer

P ∩ Q′ is just P.

5, 12

Answer: 5, 12

10
Use the P and Q diagram.
Write down the numbers that are in P′ ∪ Q′.
Show Answer

P′ ∪ Q′ is everyone except the intersection (overlap).

5, 12, 7, 11, 2, 16

Answer: 2, 5, 7, 11, 12, 16

11
Here is a Venn diagram.
Write down the numbers that are in R ∩ S′.
Show Answer

R ∩ S′ is just R.

1, 9, 14

Answer: 1, 9, 14

12
Here is a Venn diagram.
Write down the numbers that are in M′.
Show Answer

M′ is everyone not in M: just N, plus the outside.

3, 7, 11, 18

Answer: 3, 7, 11, 18

13
Here is a Venn diagram.
Write down the numbers that are in (X ∪ Y)′.
Show Answer

(X ∪ Y)′ is outside both circles: not in the union (married).

10

Answer: 10

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