(R o S) = {(x,z) | exists y s.t. (x,y) is in R and (y,z) is in S}
R-1 = {(x,y) | (y,x) is in R}
[x]R = {y in A | x R y}Quotient set: The quotient set of A w.r.t. is
[A]R = {[x]R | x is in A }
Theorem (Correspondence between equivalence relations and partitions) Given an equivalence relation R on A, [A]R is a partition of A. Viceversa, if P is a partition of A, then the relation
R = {(x,y) | exists X in P s.t. x, y are in X }is an equivalence relation on A (and the quotient of A w.r.t. R is exactly P).
(1) For every x in A, and y1, y2 in B, if both (x,y1) and (x,y2) are in f, then y1 = y2.Because of this property, we can use the more conventional notation f(x) = y for (x,y) in f.
(2) For every x in A, there exists y in B s.t. f(x) = ythen f is a total function.