Complementation (mathematics)

In set theory, the complement of a set $A$ refers to elements not in $A$ .

When all sets under consideration are considered to be subsets of a given set $U$ , the absolute complement of $A$ is the set of elements in $U$ but not in $A$ .

The relative complement of $A$ with respect to a set $B$ , also termed the difference of sets $A$ and $B$ , written $B ∖ A$ , is the set of elements in $B$ but not in $A$ .

If $A$ is a set, then the absolute complement of $A$ (or simply the complement of $A$ ) is the set of elements not in $A$ . In other words, if $U$ is the universe that contains all the elements under study, and there is no need to mention it because it is obvious and unique, then the absolute complement of $A$ is the relative complement of $A$ in $U$ :

Formally:

The absolute complement of $A$ is usually denoted by $A^{\complement }$ . Other notations include $A^{\text{c}}$ , ${\overline {A}}$ , $A'$ , $\complement _{U}A$ , and $\complement A$ .

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