Equivalence Class - Invariants

Invariants

If ~ is an equivalence relation on X, and P(x) is a property of elements of X such that whenever x ~ y, P(x) is true if P(y) is true, then the property P is said to be an invariant of ~, or well-defined under the relation ~.

A frequent particular case occurs when f is a function from X to another set Y; if f(x₁) = f(x₂) whenever x₁ ~ x₂, then f is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in the character theory of finite groups. Some authors use "compatible with ~" or just "respects ~" instead of "invariant under ~".

Any function f : X → Y itself defines an equivalence relation on X according to which x₁ ~ x₂ if and only if f(x₁) = f(x₂). The equivalence class of x is the set of all elements in X which get mapped to f(x), i.e. the class is the inverse image of f(x). This equivalence relation is known as the kernel of f.

More generally, a function may map equivalent arguments (under an equivalence relation ~_X on X) to equivalent values (under an equivalence relation ~_Y on Y). Such a function is known as a morphism from ~_X to ~_Y.