# Intersection and union of 3 sets

##### Introduction
###### Lessons
1. Introduction to Intersection and Union of 3 Sets:
2. Intersection and Union of 3 Sets
3. Principle of Inclusion and Exclusion with 3 Sets
##### Examples
###### Lessons
1. Finding Intersection and Union of 3 Sets

The Venn Diagram below shows the type of instruments that people like.

Find the following:

1. n((D$\cup$G)\B)
2. n((B$\cup$D)\G)
3. n(D$\cap$G$\cap$B)
4. n(D\G\B)
5. n((D$\cap$G)$\cup$(G$\cap$B))
2. Given the following Venn diagram:

Circle $A,B,$ and $C$ contain the same number of element. Find $a,b,$ and $c$ .

1. Richard surveyed 200 people to see which sports they like. Here is the information that Richard got:

- 70 people like soccer.

- 50 people like tennis.

- 25 people like soccer and basketball, but not tennis

- 10 people like soccer and tennis, but not basketball.

- 7 people like basketball and tennis, but not soccer

- 10 people like all three sports

How many people don't like any of the sports?

1. Principle of Inclusion and Exclusion with 3 Sets

Willy surveyed 76 people for a cake shop. Each person ate at least one of the cakes: strawberry, chocolate and vanilla. Here is the information Willy got:

- 57 ate strawberry, 50 ate chocolate, and 39 ate vanilla.

- 20 ate both strawberry and chocolate, but not vanilla.

- 15 ate strawberry and vanilla, but not chocolate.

- 5 ate chocolate and vanilla, but not strawberry.

Who ate all three types of cakes?

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###### Topic Notes

The principle of inclusion and exclusion of 3 sets says the following:

n(A$\cup$B$\cup$C) = n(A) + n(B) + n(C) - n(A$\cap$B) - n(B$\cap$C) - n(A$\cap$C) + n(A$\cap$B$\cap$C)