equivalence

An equivalence relation on a set X is a subset of X×X, i.e., a collection R of ordered pairs of elements of X, satisfying certain properties. Write "x R y" to mean (x, y) is an element of R, and we say "x is related to y, " then the properties are 1. Reflexive: a R a for all a element X, 2. Symmetric: a R b implies b R a for all a, b element X 3. Transitive: a R b and b R c imply a R c for all a, b, c element X, where these three properties are completely independent. Other notations are often used to indicate a relation, e.g., a congruent b or a~b.

equivalence class | Teichmüller space