Binary relation

A binary relation is a set of ordered pairs connecting elements of two sets, which may be the same set. Its properties depend on which pairs it contains.

Set: A collection of distinct objects treated as a single mathematical entity. A relation is itself a set, whose members are ordered pairs.
Reflexive relation: A relation on a set that contains every pair of the form (x, x). Reflexivity requires every element to be related to itself.
Equivalence relation: A reflexive, symmetric, and transitive relation on a set. It groups elements into classes of mutually equivalent objects.
Function: A mapping that assigns each input exactly one output. Unlike a general relation, a function cannot give one input multiple outputs.
Ordered pair: A pair of objects in which the first and second positions are distinguished. Each member of a binary relation records a related pair in a specific order.
Symmetric relation: A relation in which x related to y always implies y related to x. Symmetry makes every pair's reversal part of the relation.
Partial order: A reflexive, antisymmetric, and transitive relation on a set. It represents comparisons that need not make every pair comparable.
Partial function: A mapping that assigns at most one output to each input in its specified domain. It relaxes totality while retaining the single-output constraint absent from general relations.
Cartesian product: The set of all ordered pairs formed from elements of two sets. A relation from one set to another is a subset of their Cartesian product.
Antisymmetric relation: A relation in which x related to y and y related to x imply x equals y. Antisymmetry restricts which distinct elements may relate in both directions.

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