Complement Examples in Math: 42 Problems with Answers

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

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 complement of set $A$ relative to a universal set $U$ is the set of all elements in $U$ that do not belong to $A$ , written $A^c$ or $A'$ .

If the universal set is all students in your school and set $A$ is students who wear glasses, then the complement of $A$ is every student who does NOT wear glasses. It is everything outside the circle in a Venn diagram—the NOT operator applied to a set.

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 complement of A is every element of the universe U that is not in A.

Common stuck point: The procedure for complement is the easy part; the trap is computing a complement without naming the universe $U$ . Asking "Am I collecting everything in the fixed universe that is NOT in this set?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I collecting everything in the fixed universe that is NOT in this set?

Worked Examples

Example 1

easy

Let the universal set

U = \{1,2,3,4,5,6,7,8,9,10\}

and

A = \{2,4,6,8,10\}

. Find

A'

Answer

A^{'} = {1, 3, 5, 7, 9}

First step

The complement

A'

(also written

A^c

\bar{A}

) relative to universal set

U

is defined as

A' = \{x \in U : x \notin A\}

Full solution

2
List elements of $U = \{1,2,3,4,5,6,7,8,9,10\}$ that are not in $A = \{2,4,6,8,10\}$ : remove the even numbers, leaving the odd numbers.
3
Therefore $A' = \{1,3,5,7,9\}$ . Verify: $|A| + |A'| = 5 + 5 = 10 = |U|$ ✓.

The complement of a set

A

relative to the universal set

U

is everything in

U

that is not in

A

. It is essential to know the universal set.

Example 2

medium

Let

U = \{1,2,3,4,5,6,7,8,9,10\}

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

B = \{4,5,6,7\}

. Find

(A \cap B)'

Example 3

medium

Let

U = \mathbb{Z}

and

A = \{\text{even integers}\}

. Describe

A^c

Example 4

medium

Show that

A \cup A^c = U

when

A \subseteq U

Example 5

hard

In a class of

40

25

play sports. Using complements, how many do not play sports?

Practice Problems

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

Example 1

easy

Let

U = \{a,b,c,d,e,f\}

and

S = \{a,c,e\}

. Find

S'

Example 2

easy

Let the universal set be

U = \{1, 2, 3, 4, 5, 6\}

and let

A = \{2, 4, 6\}

. Find the complement of

A

Example 3

easy

With universal set

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

and

A = \{1, 2\}

, find

A^c

Example 4

easy

With

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

and

A = \{1, 3, 5\}

, find

A'

Example 5

easy

For any set

A

with universe

U

, what is

(A^c)^c

Example 6

easy

With

U = \{1,2,3,4,5,6\}

and

A = \{2,4,6\}

, how many elements are in

A^c

Example 7

easy

With

U

given, what is

U^c

Example 8

easy

With

U

given, what is

\emptyset^c

Example 9

easy

With

U = \{a,b,c,d\}

and

A = \{a,b,c,d\}

, find

A^c

Example 10

easy

With

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

and

A=\{2\}

, compute

A \cup A^c

Example 11

medium

With

U=\{1,\dots,10\}

and

A=\{2,4,6,8,10\}

, compute

A^c

and verify

|A|+|A^c|=|U|

Example 12

medium

With

U=\{1,\dots,8\}

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

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

, compute

A^c \cap B

Example 13

medium

Using De Morgan's law, rewrite

(A \cup B)^c

in terms of

A^c

and

B^c

Example 14

medium

Using De Morgan's law, rewrite

(A \cap B)^c

in terms of

A^c

and

B^c

Example 15

medium

With

U=\{1,\dots,6\}

A=\{1,2,3\}

, verify

A \cap A^c = \emptyset

Example 16

medium

|U| = 30

and

|A^c| = 12

, find

|A|

Example 17

medium

A class has 25 students; 18 passed the test. Using complement, how many did NOT pass?

Example 18

challenge

Prove De Morgan's law

(A \cup B)^c = A^c \cap B^c

Example 19

challenge

Prove that

A \subseteq B

if and only if

B^c \subseteq A^c

Example 20

challenge

With

U=\{1,\dots,20\}

, let

A

= multiples of 2 and

B

= multiples of 3. Use complement counting to find how many elements are in neither

A

nor

B

Example 21

medium

Express 'students NOT in the chess club' in set notation, with chess club

C

and universe

U

all students.

Example 22

medium

With

U=\{1,\dots,9\}

and

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

, compute

(A^c)^c

and confirm it equals

A

Example 23

easy

Let

U = \{1, 2, 3, 4, 5, 6, 7, 8\}

and

A = \{3, 5, 7\}

. Find

A^c

Example 24

medium

Let

U = \mathbb{R}

and

A = [0, 5]

. Express

A^c

in interval notation.

Example 25

medium

Let

U = \mathbb{R}

and

A = (-2, 3]

. Find

A^c

Example 26

medium

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

A = \{1, 2\}

B = \{2, 3\}

. Find

A^c \cup B^c

Example 27

medium

U = \{1, 2, 3, 4, 5, 6\}

A = \{1, 2, 3\}

B = \{3, 4\}

. Find

A^c \cap B^c

Example 28

medium

U=\{1,2,3,4,5,6,7,8,9,10\}

A=\{

even numbers in

U\}

. Find

A^c

Example 29

hard

Let

U = \mathbb{R}

. Find the complement of

A = \{x : x^2 < 4\}

Example 30

hard

U=\{1,2,\ldots,12\}

A=\{

multiples of

3

U\}

. List

A^c

Example 31

medium

U=\{a,b,c,d,e,f,g\}

A=\{a,b,c\}

B=\{c,d,e\}

. Find

(A \cup B)^c

Example 32

medium

U=\{1,2,3,4,5,6,7\}

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

B=\{3,4,5\}

. Find

(A \cap B)^c

Example 33

hard

Use De Morgan: simplify

(A \cup B)^c

Example 34

medium

U=\mathbb{R}

. What is the complement of

\{0\}

Example 35

hard

Out of

50

students,

32

take math and

28

take science. If

10

take neither, how many take math and science?

Example 36

challenge

\mathbb{R}

, find

(\mathbb{Q})^c

(the complement of the rationals).

Example 37

medium

U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}

. Find

\{x \in U : x \text{ is prime}\}^c

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Related Concepts

Set

Background Knowledge

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

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