Equivalence Relations

Abstract

Let X be a class and R a relation on X. Recall that the diagonal D _X = (x,x): x ∈ X. The relation R is said to be

• (1) reflexive if D _X + R, i.e. if for every x in X we have (x, x) ∈ R;

• (2) irreflexive ifD _X ∩ R = Ø, i.e. if for every x; in X we have (x,x) ∉ R;

• (3) symmetric if R = R^-1, i.e. if for every x, y in X such that (x, y) ∈ R we have (y, x) ∈ R;

• (4) antisymmetric if R ∩ R^-1 + D _x , i.e. if for all x, y in X such that (x,y) ∈ R and (y, x) ∈ R we have x = y.

• (5) transitive if R o R + R, i.e. if for all x, y, z in X such that (x, y) ∈ R and (y, z) ∈ R we have (x, z) ∈ R.