Complement

In set theory, the complement of a set $X$ generally refers to a set of elements which are not elements of $X$. Usually, these elements must be restricted to some set $A$ of which $X$ is a subset; in this case, we speak of the complement of $X$ with respect to $A$. Such a set is sometimes denoted $\overline{X}$, $\complement X$, $X^C$, or $X^A$.

In most standard set theories, one cannot speak of the set of all elements which are not contained in $X$, as this would imply the existance of a set of all sets, which is contradictory, as this leads to Russell's Paradox.


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