nLab
relative complement

Idea

The relative complement of $A$ in $B$ is like the complement of $A$, but relative to $B$.

Definitions

For subsets

Given two subsets $S$ and $T$ of a given ambient set $X$, the relative complement of $S$ in $T$ is the set

$$T\setminus S=\{a\mid a\in T\wedge a\notin S\}.$$T \setminus S = \{ a \;|\; a \in T \;\wedge\; a \notin S \} .

That is, $T\setminus S$ consists of those elements of $T$ that are not elements of $S$.

Note that the relative complement of $S$ in the ambient set $X$ is simply the complement $\setminus S$ of $S$; sometimes this is called the absolute complement. Conversely, $T\setminus S$ is the intersection of $T$ and $\setminus S$.

In material set theory

In material set theory, all sets are subsets of the proper class $V$ of all sets (or all small things, if we might have atoms). In this case, there is no absolute complement (except as another proper class), but we can still form the relative complement:

$$T\setminus S=\{a\in T\mid a\notin S\}.$$T \setminus S = \{ a \in T \;|\; a \notin S \} .

In a lattice

Given a lattice $L$ and two elements $x$ and $y$ of $L$, a relative complement of $x$ in $y$ in an element $y\setminus x$ such that:

• $x\wedge (y\setminus x)=\perp$ (where $\wedge$ is the meet in the lattice and $\perp$ is the bottom of the lattice) and
• $y\le x\vee (y\setminus x)$ (where $\vee$ is the join in the lattice).

The term Boolean ring is sometimes used for something like a Boolean algebra in which there may be no top element (and therefore no complements) but still relative complements; compare the notions of ring vs algebra at sigma-algebra.

This includes the previous examples; the power set of $X$ is a Boolean algebra, while the class of all sets is (in this sense) a Boolean ring.