Equality relation

A relation on a set that relates each element only to itself. It is reflexive, symmetric, and transitive.

Binary relation: A set of ordered pairs connecting elements of two sets, or of one set to itself. An equality relation is a particular binary relation on a set.
Diagonal relation: The relation on a set whose ordered pairs have identical first and second entries. The equality relation is precisely the diagonal relation on its underlying set.
Equivalence relation: A reflexive, symmetric, and transitive relation on a set. Every equality relation is an equivalence relation, but others can relate distinct elements.
Substitution property of equality: The rule that equal expressions may replace one another in any statement without changing its truth. It is the central inference rule licensed by equality in mathematical reasoning.
Ordered pair: A pair of objects whose identity depends on both entries and their order. A relation includes ordered pairs, with equality containing precisely the pairs (x, x).
Identity function: A function from a set to itself that maps every element to itself. Its graph consists of exactly the ordered pairs in the equality relation.
Congruence relation: An equivalence relation compatible with the operations of an algebraic structure. Congruence can identify distinct elements while preserving structure; equality cannot.
First-order logic with equality: First-order logic extended with a symbol interpreted as identity across the domain. Its axioms and rules formalize reasoning with equality relations.
Reflexive relation: A relation on a set in which every element is related to itself. Equality satisfies reflexivity because every element equals itself.
Equality symbol: The mathematical symbol =, used to assert that two expressions denote the same object. The symbol expresses the relation's membership test in ordinary mathematical notation.

Linked from 6 pages