Practice Intersection in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The intersection of sets $A$ and $B$ is the set of all elements that belong to both $A$ and $B$ simultaneously, written $A \cap B$ .

Picture two overlapping circles in a Venn diagram—the intersection is only the overlapping region where both circles cover. For example, if set $A$ is students who play soccer and set $B$ is students who play piano, then $A \cap B$ is students who do both. It is the AND gate of set theory: an element must satisfy both conditions to be included.

Showing a random 20 of 50 problems.

Example 1

medium

Compute

A \cap B

where

A = \{1, 2, 3, 4, 5\}

and

B

is the set of odd numbers in

A

Example 2

medium

Let

A = \{x \in \mathbb{R} : x \ge 1\}

and

B = \{x \in \mathbb{R} : x \le 4\}

. Find

A \cap B

Example 3

medium

Compute

(-\infty, 3) \cap [0, \infty)

Example 4

challenge

Sets

A, B, C

are pairwise disjoint with

|A|=3, |B|=4, |C|=5

. Find

|A \cup B \cup C|

and

|A \cap B \cap C|

Example 5

easy

Let

M = \{2, 4, 6, 8\}

and

N = \{1, 2, 3, 4\}

. Find

M \cap N

Example 6

medium

Let

E

= even integers,

P

= positive integers

\le 10

. Find

E \cap P

Example 7

easy

True or false:

A \cap \emptyset = A

Example 8

easy

Find

A \cap B

A = \{2, 4, 6, 8\}

and

B = \{1, 2, 3, 4, 5\}

Example 9

hard

Compute

\bigcap_{n=1}^{\infty} \left[0, \tfrac{1}{n}\right]

Example 10

medium

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

. What property is this?

Example 11

hard

Prove

A \cap (A \cup B) = A

(absorption).

Example 12

hard

De Morgan: rewrite

(A \cap B)^c

in terms of complements and union.

Example 13

medium

List the intersection of 'multiples of 2' and 'multiples of 5' up to 20.

Example 14

easy

Compute

\{1, 2\} \cap \{3, 4\}

Example 15

medium

In a survey, 20 like coffee, 15 like tea, and 25 like at least one. How many like both?

Example 16

easy

Compute

A \cap A

for

A = \{5, 6\}

Example 17

easy

Compute

\{2, 4, 6\} \cap \{1, 3, 5\}

Example 18

easy

How many elements are in

\{2,4,6,8,10\} \cap \{1,2,3,4,5\}

Example 19

easy

Let

A = \{1, 2, 3, 4\}

and

B = \{3, 4, 5, 6\}

. Find

A \cap B

Example 20

easy

How many elements are in

\{1, 2, 3, 4\} \cap \{2, 4, 6\}

← Back to Intersection Worked Examples Formula Explained

Related Concepts

Set Union Venn Diagram