nLab
relative complement

Idea

The relative complement of AA in BB is like the complement of AA, but relative to BB.

Definitions

For subsets

Given two subsets SS and TT of a given ambient set XX, the relative complement of SS in TT is the set

TS={aaTaS}. T \setminus S = \{ a \;|\; a \in T \;\wedge\; a \notin S \} .

That is, TST \setminus S consists of those elements of TT that are not elements of SS.

Note that the relative complement of SS in the ambient set XX is simply the complement S\setminus S of SS; sometimes this is called the absolute complement. Conversely, TST \setminus S is the intersection of TT and S\setminus S.

In material set theory

In material set theory, all sets are subsets of the proper class VV 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:

TS={aTaS}. T \setminus S = \{ a \in T \;|\; a \notin S \} .

In a lattice

Given a lattice LL and two elements xx and yy of LL, a relative complement of xx in yy in an element yxy \setminus x such that:

  • x(yx)= x \wedge (y \setminus x) = \bot (where \wedge is the meet in the lattice and \bot is the bottom of the lattice) and
  • yx(yx) y \leq 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 XX is a Boolean algebra, while the class of all sets is (in this sense) a Boolean ring.

Revised on December 16, 2009 21:02:48 by Toby Bartels (173.60.119.197)