Question

I. In each of the flbwing prdblems, the relation Bis defined in the set Z of all the integers. Say in eadh case if Ris: ne Reflexive Symmetnic 3 Antisymmetnic Transt ve Partial arder relotian 6) Equivalence relotion Justfy yaur a.xRy fondonly if x-2y b.xRy if ond only if X=-y c. xRy ifond only f X <Y d.xRy ifond anly if x2y e. xRy Ff and only if x-y-sk Pa any kez S onswer:

Answer 1

a) xRy Ris depined on the ct all integors Z A nelation R is Baud t re naexve Rz Symmettue R Rx Thram&e tit 42RY, YRZ→スRZ Pantal &der a RyRx=y yy md yRa R Epuivalene tatati om Ratleive, ai Symmetruc and C amsve とてキ xEZ Sincs Ry x= 2 is mt neflexive xRy う 23 えこ 2 ) 2 (a), ymmetru c met Rの 24 つしこ bw me ze d az Rz 3 24 = 2 ( 2z ) Ry x 21
8
met aeiv, mct mmetruc, mt thamatir partal ade neletren Hence it mct a amd mot ealence redaten NxeZ Rin mt ro uve xR -4 )(Bymult with - YR R semmetrui 介り

Let Ry YRz Na comder (2-) Rn mct tramative Sime Rin lexiv, mmetuc bt met tramie, R mt der a tralation and mct an euva lence nelation RY eZ 2 R met ove RY k Ru met &mmet ru e Nad Rz ppt ) (4 Z Rz yRz :. R us tramsive

Ris mct nfleve mt 4mnetrue Sinue t tramsdive, ili et am ralation, m ulonce dor ralatim d) Simue eZ(Lacou at Rx Ris rooile x Ry Rx Hence t nt mmetnue but Rx ant mmet ruc Naw Ry 178 4 Rz z R the 7Z Rz :Rus tramalire Since Rn nofloxivt, amti Aemmchruc and tramtve but it is mtam euvalonce nolati om Rna panti dor nelatim

RY sk 5(0) Rx xe rofle Rin k ラー(9-) こ 5k 5-k) Ry yR ,h-keZ y-ズ ニ Ro mmatue Flow Comorden, つ Ry,→スーヤに5k Rz y-Z 5L Z Comden 5 (k+) ラ Rz unty Rs natexir, mmeuc and tamtivr hence it eaualomce aleton