Complement (set theory)

In set theory, the complement of a set A, often denoted by

A

c

{\displaystyle A^{c}}

(or A′), is the set of elements not in A.

When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set U, the absolute complement of A is the set of elements in U that are not in A.

The relative complement of A with respect to a set B, also termed the set difference of B and A, written

B

∖

A

,

{\displaystyle B\setminus A,}

is the set of elements in B that are not in A.

Absolute complement

Definition

If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the relative complement of A in U:

A

c

=

U

∖

A

=

{

x

∈

U

:

x

∉

A

}

.

{\displaystyle A^{c}=U\setminus A=\{x\in U:x\notin A\}.}

The absolute complement of A is usually denoted by

A

c

{\displaystyle A^{c}}

. Other notations include

A

¯

{\displaystyle {\overline {A}}}

,

A

′

,

{\displaystyle A',}

∁

U

A

,

and

∁

A

.

{\displaystyle \complement _{U}A,{\text{ and }}\complement A.}

Examples

Assume that the universe is the set of integers. If A is the set of odd numbers, then the complement of A is the set of even numbers. If B is the set of multiples of 3, then the complement of B is the set of numbers congruent to 1 or 2 modulo 3 (or, in simpler terms, the integers that are not multiples of 3).

Assume that the universe is the standard 52-card deck. If the set A is the suit of spades, then the complement of A is the union of the suits of clubs, diamonds, and hearts.