1.Consider a binary relationRonAwhich is reflexive and transitive. Define binary relationsE_RandP_RonAas follows.

x,yER(x,yR y,xR)

and

x,yER(x,yR y,xR)

(a) Prove thatE_Ais an equivalence relation onA.

(b) Prove thatP_Ais a strict partial ordering onA.

2.Consider the following binary relations. In each case prove the relation in question is an equivalence relation and describe, in geometric terms, what the equivalence classes are.

(a)S1is a binary relation onR²R²defined by

x,y,x,yS1 |x| +| y|=|x| + |y|

Recall thatR²=RR.

(b)S2is a binary relation onRdefined by

x,yS2 xy Z (Interger)