Editing Binary relation (section)

== Sets versus classes ==
Certain mathematical "relations", such as "equal to", "subset of", and "member of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of [[axiomatic set theory]]. For example, to model the general concept of "equality" as a binary relation <math>=</math>, take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory.

In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem is to select a "large enough" set <math>A</math>, that contains all the objects of interest, and work with the restriction <math>=_A</math> instead of <math>=</math>. Similarly, the "subset of" relation <math>\subseteq</math> needs to be restricted to have domain and codomain <math>P(A)</math> (the power set of a specific set <math>A</math>): the resulting set relation can be denoted by <math>\subseteq_A.</math> Also, the "member of" relation needs to be restricted to have domain <math>A</math> and codomain <math>P(A)</math> to obtain a binary relation <math>\in_A</math> that is a set. [[Bertrand Russell]] has shown that assuming <math>\in</math> to be defined over all sets leads to a contradiction in [[naive set theory]], see ''[[Russell's paradox]]''.

Another solution to this problem is to use a set theory with proper classes, such as [[Von Neumann–Bernays–Gödel set theory|NBG]] or [[Morse–Kelley set theory]], and allow the domain and codomain (and so the graph) to be [[proper class]]es: in such a theory, equality, membership, and subset are binary relations without special comment. (A minor modification needs to be made to the concept of the ordered triple <math>(X, Y, G)</math>, as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the binary relation with its graph in this context.)<ref>{{cite book |title=A formalization of set theory without variables |last1=Tarski |first1=Alfred |author-link=Alfred Tarski |last2=Givant |first2=Steven |year=1987 |page=[https://archive.org/details/formalizationofs0000tars/page/3 3] |publisher=American Mathematical Society |isbn=0-8218-1041-3 |url=https://archive.org/details/formalizationofs0000tars/page/3 }}</ref> With this definition one can for instance define a binary relation over every set and its power set.