nLab complement

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Contents

Idea

In sets

The complement XSX \setminus S of a subset SXS \subset X of a set XX is the set of ell elements of XX not contained in SS:

XS{xX|xS}. X \setminus S \;\coloneqq\; \big\{ x \in X \;|\; x \notin S \big\} \,.

(Besides XSX \setminus S, there are many other notations, such as XSX - S, S¯\bar{S}, S cS^c, ¬S\neg{S}, and so forth.)

Notice that S(XS)=S \cap (X \setminus S) = \empty, while S(XS)=XS \cup (X \setminus S) = X by the principle of excluded middle.

In lattices

A complement of an element SS of a lattice is an element TT such that ST=S \wedge T = \bot and ST=S \vee T = \top. Note that in general, complements need not be unique; for example, in the lattice of vector subspaces of a 2-dimensional vector space over a field kk, a 1-dimensional subspace will have as many complements as there are elements of kk. However, in some cases complements will be unique, for example in a distributive lattice, in which case it is denoted S˜\tilde{S} (or ¬S\neg S, etc.).

If every element has a complement, one speaks of a complemented lattice. Examples are Boolean algebras, and in fact complemented distributive lattices are the same thing as Boolean algebras (in the sense that the category of Boolean algebras is equivalent to the category of complemented distributive lattices).

More generally, the pseudocomplement of an element SS of a Heyting algebra is given by S˜=S\tilde{S} = S \Rightarrow \bot. This satisfies SS˜=S \wedge \tilde{S} = \bot but not SS˜=S \vee \tilde{S} = \top in general. This case includes the complement of a subset even in constructive mathematics.

In any category

In another direction, the complement of a complemented subobject SS of an object XX in a coherent category is the unique subobject S˜\tilde{S} such that SS˜S \cap \tilde{S} is the initial object and SS˜=XS \cup \tilde{S} = X.

In truth values

The complement of a truth value (seen as a subset of the point) is called its negation.

 Strict complements

In constructive mathematics, the denial inequality is not a tight apartness relation, and sets in general do not have a tight apartness relation. This means that every set with a tight apartness relation has two notions of inequality, the normal denial inequality \neq and the tight apartness relation #\#. Thus, there are two notions of complement. There is the usual notion of complement, where given a set XX, every element in the complement of a subset SXS \subseteq X is not equal to any element in SS. Then there are strict complements, where given a set XX with a tight inequality relation #\#, every element in the strict complement of a subset SXS \subseteq X is apart from every element in SS.

Last revised on December 9, 2022 at 03:30:23. See the history of this page for a list of all contributions to it.