Definition 3

Let $\mathrm{A}$ be a subset of a set $\mathrm{X}$. The set of elements of $\mathrm{X}$ which do not belong to $\mathrm{A}$, that is to say the set $\mathscr{E}x$ $(x \notin \mathrm{A}$ and $x \in \mathrm{X})$, is called the complement of $\mathrm{A}$ in $\mathrm{X}$, and is denoted by $\complement_{\mathrm{X}}\mathrm{A}$ or $\mathrm{X} - \mathrm{A}$ (or by $\complement\mathrm{A}$ if there is no risk of confusion).

Let A be a subset of a set X; the relations "$x \in \mathrm{X}$ and $x \notin \mathrm{A}$" and $x \in \complement_{\mathrm{X}}\mathrm{A}$ are then equivalent. Consequently the relation "$x \in \mathrm{X}$ and $x \notin \complement_{\mathrm{X}}\mathrm{A}$" is equivalent to "$x \in \mathrm{X}$ and $(x \notin \mathrm{X}$ or $x \in \mathrm{A})$", hence to $x \in A$. In other words, $A = \complement_X(\complement_X A)$ is a true relation. Similarly, one shows that if B is a subset of X, the relations $A \subset B$ and $\complement_X B \subset \complement_X A$ are equivalent.