Complement (set theory)

In set theory, the complement of a set $A$ is the set of all elements that are not in $A$ .

Absolute complement

If the universal set $U$ is defined, then, in terms of the relative complement, the complement of $A$ is

$A^C=U\setminus A$

Relative complement

Given two sets $A, B$ , the relative complement of $A$ in $B$ , denoted as $A \setminus B$ (sometimes $A-B$) is

$A\setminus B=\{x|x\in B\and x\in A\}$

Example

1. Suppose the universe $U$ is the set of all letters in the English alphabet. The complement of the set of all vowels in $U$ , $V^C$ ($V$ being the set), is the set of all consonants.

See also

• Negation (equivalent in logic)
• Intersection (set theory)
• Union
• Symmetric difference