Complement (set theory)

In set theory, the complement of a set $A$ refers to elements 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 $A$ . 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 $A$ .

If $A$ and $B$ are sets, then the relative complement of $A$ in $B$ , also termed the set-theoretic difference of $B$ and $A$ , is the set of elements in $B$ but not in $A$ .

The relative complement of $A$ in $B$ is denoted $B ∖ A$ according to the ISO 31-11 standard. It is sometimes written $B - A$ , but this notation is ambiguous, as in some contexts it can be interpreted as the set of all elements $b - a$ , where $b$ is taken from $B$ and $a$ from $A$ .

Formally:

Let $A$ , $B$ , and $C$ be three sets. The following identities capture notable properties of relative complements:

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 sets 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 $U$ :

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