Equivalence Class - Notation and Formal Definition

Notation and Formal Definition

An equivalence relation is a binary relation ~ satisfying three properties:

• For every element a in X, a ~ a (reflexivity),
• For every two elements a and b in X, if a ~ b, then b ~ a (symmetry)
• For every three elements a, b, and c in X, if a ~ b and b ~ c, then a ~ c (transitivity).

The equivalence class of an element a is denoted and may be defined as the set

of elements that are related to a by ~. The alternative notation _R can be used to denote the equivalence class of the element a specifically with respect to the equivalence relation R. This is said to be the R-equivalence class of a.

The set of all equivalence classes in X given an equivalence relation ~ is denoted as X/~ and called the quotient set of X by ~. Each equivalence relation has a canonical projection map, the surjective function π from X to X/~ given by π(x) = .