Setoid

• A setoid is a set, S, equipped with an equivalence relation (~) and denoted by (S, ~).

• setoid = $(S, \thicksim )$

• A Setoid may also be called E-set, Bishop set, or extensional set.

• Setoids are studied especially in proof theory and in type-theoretic foundations of mathematics.

Frequently, defining an equivalence relation on a set is immediately followed by the formation of the quotient set, by turning the equivalence into equality.

In contrast, setoids come handy when difference between identity and equivalence must be maintained, which is often the case with interpretation of intensional equality (the equality on the original set) and extensional equality (the equivalence relation, or the equality on the quotient set).