# Complementation

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An operation which brings a subset $M$ of a given set $X$ into correspondence with another subset $N$ so that if $M$ and $N$ are known, it is possible in some way to reproduce $X$. Depending on the structure with which $X$ is endowed, one distinguishes various definitions of complementation, as well as various methods of reconstituting $X$ from $M$ and $N$.

## Contents

### Sets

In the general theory of sets the complement of a subset $M$ (or complementary subset, relative complement) in a set $X$ is the subset $\complement_X M$ (or $\complement M$ if $X$ is assumed, or $X \setminus M$) consisting of all elements $x \in X$ not belonging to $M$; an important property is the duality principle (one of the De Morgan laws): $$\complement \bigcup_{\xi} M_\xi = \bigcap_\xi \complement M_\xi$$

### Lattices

Let $L$ be a lattice with 0 and 1and $n$ an element of $L$ Then $m$ is a complement of $n$ if $m \vee n = 1$, $m \wedge n = 0$. In a complemented lattice each element has at least one complement; a distributive lattice has the property that each element has at most one complement. A Boolean lattice is a distributive lattice in which each element has a (unique) complement.

How to Cite This Entry:
Complementation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complementation&oldid=42540
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article