Equivalence relations

A reflexive, symmetric and transitive relation partitions a set into equivalence classes.

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Order relations
A partial order is reflexive, antisymmetric and transitive; not every pair must be comparable.
Relations
A binary relation is a subset of a Cartesian product; its properties constrain related pairs.
Axiomatic systems
An axiomatic theory derives theorems from stated axioms and inference rules.
Bijections
A bijection is both injective and surjective and has a two-sided inverse.
Cartesian products
A Cartesian product consists of ordered pairs with one component from each set.