Basic Algebra B.2 Set Theoretic Basics

Question 1

binary relation between sets

Answer 1

Definition. A (binary) relation between sets X and Y is a subset S of the Cartesian product X × Y. If X = Y, then we call S a relation on X. The notation xSy is often used to denote the fact that (x,y) € S.

Question 2

domain, range, one-to-one and onto

Answer 2

Definition. Let S be a relation between sets X and Y. Define subsets dom(S) and range(S), called the domain and range of S, respectively, by

dom(S) = {x| xSy for some y in Y}

and

range(S) ={y| xSy for some x in X}.

The relation S is said to be one-to-one if x1Sy and x2Sy imply that x1 = x2. The relation S is said to be onto Y if for all y in Y there is an x in X so that xSy. S is said to be well defined if xSy1 and xSy2 imply that y1 = y2.