Union Examples in Math

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

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

Concept 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.

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: The union gathers all elements in A, in B, or in both, into one set.

Common stuck point: The procedure for union is the easy part; the trap is writing a shared element twice in the union. Asking "Does an item belong as long as it is in at least one of the sets?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does an item belong as long as it is in at least one of the sets?

Worked Examples

Example 1

easy

Let

A = \{1, 3, 5\}

and

B = \{2, 3, 4\}

. Find

A \cup B

Answer

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

First step

Recall the definition:

A \cup B = \{x : x \in A \text{ or } x \in B\}

. The word 'or' is inclusive — an element belongs to the union if it appears in at least one of the sets.

Full solution

2
List all elements from $A = \{1,3,5\}$ and $B = \{2,3,4\}$ , including each at most once: 1 (from $A$ ), 2 (from $B$ ), 3 (in both), 4 (from $B$ ), 5 (from $A$ ).
3
Therefore $A \cup B = \{1,2,3,4,5\}$ . Notice $|A \cup B| = 5 = |A| + |B| - |A \cap B| = 3 + 3 - 1$ , confirming the inclusion-exclusion principle.

The union operator collects all elements from both sets. Duplicate elements are listed only once because sets contain distinct elements.

Example 2

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 3

medium

A

and

B

are disjoint with

|A| = 5

and

|B| = 7

, find

|A \cup B|

Example 4

hard

Let

A_n = \{1, 2, \ldots, n\}

. Describe

A_3 \cup A_5 \cup A_2

Practice Problems

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

Example 1

easy

Let

P = \{a, b, c\}

and

Q = \{c, d, e\}

. Find

P \cup Q

Example 2

easy

Find

A \cup B

A = \{a, b, d\}

and

B = \{b, c, d, e\}

Example 3

easy

Compute

A \cup B

for

A = \{1, 2, 3\}

and

B = \{2, 3, 4\}

Example 4

easy

Compute

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

Example 5

easy

Compute

A \cup \emptyset

for

A = \{5, 6\}

Example 6

easy

Compute

A \cup A

for

A = \{7, 8\}

Example 7

easy

How many elements are in

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

Example 8

easy

Compute

\{a, b, c\} \cup \{c\}

Example 9

easy

Compute

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

Example 10

easy

A \subseteq A \cup B

always true?

Example 11

medium

Compute

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

Example 12

medium

|A| = 5

|B| = 3

, and

|A \cap B| = 2

, find

|A \cup B|

Example 13

medium

In a class, 12 students play soccer and 8 play tennis; 5 play both. How many play at least one sport?

Example 14

medium

Compute

A \cup B

where

A = \{x : 1 \le x \le 3, x \in \mathbb{Z}\}

and

B = \{x : 2 \le x \le 5, x \in \mathbb{Z}\}

Example 15

medium

A \cup B = B

, what relationship must hold between

A

and

B

Example 16

medium

Compute the union of

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

(all four sets together).

Example 17

medium

A = \{1, 2, 3\}

and

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

, what is the smallest possible

B

Example 18

challenge

Prove that union is commutative:

A \cup B = B \cup A

Example 19

challenge

Prove the inclusion-exclusion formula

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

for finite sets.

Example 20

challenge

Three sets satisfy

|A|=|B|=|C|=10

, each pairwise intersection has size 4, and

|A\cap B\cap C|=2

. Find

|A\cup B\cup C|

Example 21

medium

Express 'students who like math OR science' in set notation, given math-likers

M

and science-likers

S

Example 22

medium

Express 'numbers that are prime OR even, up to 6' as a union and list it.

Example 23

easy

Let

A = \{1, 4, 7\}

and

B = \{2, 4, 6\}

. Find

A \cup B

Example 24

medium

|A| = 6

|B| = 9

, and

|A \cap B| = 2

, find

|A \cup B|

Example 25

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 26

medium

Let

A = (-\infty, 3]

and

B = [1, 5)

. Express

A \cup B

in interval notation.

Example 27

medium

Let

A = (-2, 1)

and

B = (3, 7)

. Express

A \cup B

Example 28

easy

Let

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

and

A = \{2, 4\}

. Find

A \cup U

Example 29

hard

In a class of

30

students,

18

play soccer,

12

play basketball, and

5

play both. How many play at least one?

Example 30

hard

Sets

A

and

B

satisfy

|A|=10

|B|=14

|A \cup B|=20

. Find

|A \cap B|

Example 31

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 32

easy

Compute

A \cup B

for

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

and

B = \{1, 3, 5\}

Example 33

medium

Let

A = \{

multiples of

3

from

1

20\}

and

B = \{

multiples of

5

from

1

20\}

. List

A \cup B

Example 34

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 35

hard

In the survey above, how many like both coffee and tea?

Example 36

medium

List

A \cup B \cup C

for

A=\{1,2\}

B=\{2,3\}

C=\{3,4\}

Example 37

challenge

In a survey of

200

people:

120

read book A,

100

read book B,

80

read book C,

50

read A and B,

40

read B and C,

30

read A and C,

20

read all three. How many read at least one?

Example 38

medium

A \cup B = A

, what must be true about

B

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Related Concepts

Set Intersection Venn Diagram

Background Knowledge

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

set