Equivalency

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:

As a consequence of the reflexive, symmetric, and transitive properties, any equivalence relation provides a partition of the underlying set into disjoint equivalence classes.

Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a ~ b" and "a ≡ b", which are used when R is implicit, and variations of "a ~_Rb", "a ≡_Rb", or "aRb" to specify R explicitly. Non-equivalence may be written "a ≁ b" or "a ≢ b".

A given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. That is, for all a, b and c in X:

X together with the relation ~ is called a setoid. The equivalence class of ${\displaystyle a}$ under ~, denoted ${\displaystyle [a]}$, is defined as ${\displaystyle [a]=\{b\in X\mid a\sim b\}}$.

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