Practice Complement 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 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.

Showing a random 20 of 50 problems.

Example 1

hard

Let

U = \mathbb{R}

. Find the complement of

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

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

Let

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

and

S = \{a,c,e\}

. Find

S'

Example 4

easy

With

U

given, what is

U^c

Example 5

medium

|U| = 30

and

|A^c| = 12

, find

|A|

Example 6

easy

Let

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

and

A = \{3, 5, 7\}

. Find

A^c

Example 7

easy

Compute

A \cap A^c

for any set

A

in a universe

U

Example 8

easy

With universal set

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

and

A = \{1, 2\}

, find

A^c

Example 9

medium

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

A = \{1, 2\}

B = \{2, 3\}

. Find

A^c \cup B^c

Example 10

medium

Show that

A \cup A^c = U

when

A \subseteq U

Example 11

medium

With

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

A=\{1,2,3\}

, verify

A \cap A^c = \emptyset

Example 12

challenge

\mathbb{R}

, find

(\mathbb{Q})^c

(the complement of the rationals).

Example 13

medium

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

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

B=\{3,4,5\}

. Find

(A \cap B)^c

Example 14

medium

Let

U = \mathbb{Z}

and

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

. Describe

A^c

Example 15

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 16

challenge

Prove that

A \subseteq B

if and only if

B^c \subseteq A^c

Example 17

medium

With

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

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

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

, compute

A^c \cap B

Example 18

medium

Using De Morgan's law, rewrite

(A \cap B)^c

in terms of

A^c

and

B^c

Example 19

easy

With

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

and

A = \{2,4,6\}

, how many elements are in

A^c

Example 20

medium

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

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