Editing Equivalence relation (section)

== Well-definedness under an equivalence relation ==

If <math>\,\sim\,</math> is an equivalence relation on <math>X,</math> and <math>P(x)</math> is a property of elements of <math>X,</math> such that whenever <math>x \sim y,</math> <math>P(x)</math> is true if <math>P(y)</math> is true, then the property <math>P</math> is said to be [[well-defined]] or a {{em|class invariant}} under the relation <math>\,\sim.</math> 

A frequent particular case occurs when <math>f</math> is a function from <math>X</math> to another set <math>Y;</math> if <math>x_1 \sim x_2</math> implies <math>f\left(x_1\right) = f\left(x_2\right)</math> then <math>f</math> is said to be a {{em|morphism}} for <math>\,\sim,</math> a {{em|class invariant under}} <math>\,\sim,</math> or simply {{em|invariant under}} <math>\,\sim.</math> This occurs, e.g. in the character theory of finite groups. The latter case with the function <math>f</math> can be expressed by a commutative triangle. See also [[Invariant (mathematics)|invariant]].  Some authors use "compatible with <math>\,\sim</math>" or just "respects <math>\,\sim</math>" instead of "invariant under <math>\,\sim</math>".

More generally, a function may map equivalent arguments (under an equivalence relation <math>\,\sim_A</math>) to equivalent values (under an equivalence relation <math>\,\sim_B</math>).  Such a function is known as a morphism from <math>\,\sim_A</math> to <math>\,\sim_B.</math>