Question

(i) Prove that the realtion in Z of congruence modulo p is an equivalence relation. Namesly, show that Rp := {(a,b) € ZxZ:a = 5(p)} is reflexive, symmetric and transitive. (ii) Let pe N be fixed. Show that there are exactly p equivalence classes induced by Rp. (iii) Consider the relation S E N defined as: a Sb if and only if a b( i.e., a divides b). Prove that S is an order relation. In other words, S := {(a,b) € Nx N: a b} is reflexive, anti-symmetric and transitive. Reflexive: (iv) Are every two integer numbers related by the order defined in (iii)? Give a short proof if the answer yes. If the answer is no, supply an example. It's not related by the order. Let a = 2 and b= -2. Then, abas -2-(-1). 2 and 6 a as 2 = (-1)-(-2). Hence a Rb and bRa, but a + b which breaks anti-symmetric.

Answer 1

seludims a Rp = {(arb) & 7x7 as b (modes} For every act elo = ara ?) as a modo > (@a) € Rp - Se Re is reeflerine, het (a,b) € Rp 2) as b mode a planb. - Alb- al 7 be a modo (b, a) ERR So Cabl € Rp y (ra) tlp. a By B symmetric, - let (a,b) é Rp and (do, c) Elp & asb mode and b= c mode. 2 pl arb and plb-c a pl (a-b) +(5.c) = arc a = c modo (ac) ERR So (a,b), (.b,c) ERp 2) (ac) ERO » Rp is Aransitive, fresca 0, 0, 0 pp ei egociolence reelation- (ii) a E {0,112,...10.13 then - class of a = [a] = { bezlash moda? So there are e equivalence classes induced by a - lis SE { cable in 'XIN' alb)__ taEn alla =) ala, So s es reflexive, 6. If all and bra then b=ta. . but ab En implies asb So şes Antigymmetrica

let alb and blc. 7 lim such that adab and c=bm Now Coloma alm - all. So (a,b) ES and Cebec) ES => [acc) € S. - So s is Areansitive . from 0, 0, s es paretial orden relation, fix) Every two entegers are not related let as2, b=3. Then neither Cables ore (bra)ES-