Relations and maps: structuring sets

A notion of relation between sets is essential if we are to begin doing mathematics. At the heart of the foundations of mathematics, the concept of a relation includes maps as a special case and gives sets structure.

Written by

FRANCOIS LAVALLOU

PublishedOctober 6, 2016UpdatedMay 16, 20261 min read

Three distinct foundations of mathematics can be identified. A logical foundation consists in establishing the consistency of mathematics as a whole. It must never be possible for two valid proofs to establish that the same proposition is both true and false. An axiomatic foundation begins with a finite set of axioms from which all of mathematics must be deducible using the rules of inference of a given logic. This approach corresponds to the framework of set theory.

The conceptual foundation of mathematics, represented by category theory, identifies the fundamental concepts and the links between them, thereby providing the grammar and syntax of a universal language in which all of mathematics can be conceived. The underlying approach embodies the purest mathematical spirit: focusing on substance rather than form—a group, then, is more a structure than a set! The latter two approaches are closely related: category theory can serve as an axiomatic foundation, and set theory as a conceptual one. This is why the notion of relation, which is inherent in set theory and closely linked to the notion of structure, plays a central role in mathematics.

Relations and ordered pairs --------------------

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