Practice Union 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 union of sets $A$ and $B$ is the set of all elements that belong to $A$ , to $B$ , or to both, written $A \cup B$ .

Pour both sets into one container and remove duplicates. Everything from either pile ends up in the union — this is the OR operation for sets.

Showing a random 20 of 50 problems.

Example 1

easy

Let

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

and

A = \{2, 4\}

. Find

A \cup U

Example 2

easy

Compute

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

Example 3

hard

Let

A = \{x \in \mathbb{R} : x^2 \le 9\}

and

B = \{x \in \mathbb{R} : x \ge 2\}

. Express

A \cup B

Example 4

easy

Let

P = \{a, b, c\}

and

Q = \{c, d, e\}

. Find

P \cup Q

Example 5

easy

Compute

\emptyset \cup \{4, 5, 6\}

Example 6

medium

Let

A = \{x \in \mathbb{R} : x > 2\}

and

B = \{x \in \mathbb{R} : x < 5\}

. Express

A \cup B

in interval notation.

Example 7

medium

Let

A = \{x \in \mathbb{Z} : -2 \le x \le 2\}

and

B = \{x \in \mathbb{Z} : 0 \le x \le 4\}

. Find

A \cup B

Example 8

medium

Compute the union of

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

(all four sets together).

Example 9

easy

A \subseteq A \cup B

always true?

Example 10

hard

In a class of

30

students,

18

play soccer,

12

play basketball, and

5

play both. How many play at least one?

Example 11

easy

B \subseteq A \cup B

always true?

Example 12

medium

Let

A = \{

multiples of

3

from

1

20\}

and

B = \{

multiples of

5

from

1

20\}

. List

A \cup B

Example 13

easy

Compute

\{1, 2\} \cup \{2, 1\}

Example 14

medium

|A| = 5

|B| = 3

, and

|A \cap B| = 2

, find

|A \cup B|

Example 15

medium

True or false: for any sets

A, B

A \cup B = B \cup A

Example 16

hard

60

people are surveyed:

40

like coffee,

35

like tea, and

5

like neither. How many like at least one of coffee or tea?

Example 17

easy

Compute

A \cup \emptyset

for

A = \{5, 6\}

Example 18

easy

Compute

A \cup B

for

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

and

B = \{1, 3, 5\}

Example 19

challenge

Prove the inclusion-exclusion formula

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

for finite sets.

Example 20

medium

List

A \cup B \cup C

for

A=\{1,2\}

B=\{2,3\}

C=\{3,4\}

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Related Concepts

Set Intersection Venn Diagram