Question

Define a relation < on Z by m <n iff |m| < |n| or (\m| = |n| 1 m <n) (a) Prove that < is a partial order on Z. (b) A partial order R on a set S is called a total order (or linear order) iff (Vx, Y ES)(x + y + ((x, y) E R V (y,x) E R)) Prove that is a total order on Z. (c) List the following elements in <-increasing order. –5, 2, 0, -3, 1, -2

Answer 1

Define Relation a by Im n ff m n Def Pavdial ordor Realon On Relation A on a aled pardtial orden Reaton Reflexiue Antisymmetie tanstive (3 ) Partial orde is On Z Proue that Relation Reflenive - meZ For m gm ms [m{ A So Reflexive i's aa So Relation

Antsymmetie * . bs a a nd a gb then a b a b alb A asb -L OR (a121bl and -T \bHa Þsa b a OR ab 41-(b nca< n lbl < 91 not possible 2 a1(81 \b\ = [9{ ^ NOT Posible (e1b (b1< (a| not possible ab ba uhen ab A Se antrsymmtue g1uen Relaton So (3) Transtue Casb ond ) lal= b ab a b O R fa1<Ib

As weha Seen -tnther a a1lb n a b A \blE (C\ n b<c a Lc a c (91= (b A a zb ^ \bl< \C\ \91くIC a C T and (a1 (b b= (C| nb<C へ 91 21c1 a C ana al te tranitiue partial Order Scf 3Iuen Relalon So IS a.

(b Total ordler (z,4)-patial order 362) X6J or OR 191 = [X{ OR 1911x1 For any tuo ReQ numbuy For any ,J EZ Always hau we usunD Cqual Sgn usval USuap ( 13 m Both ave Both -ve (9] Z (3] one -ve One Ave oR Same Procegs oR No Need to include Because in Dep ab Tofad Order x¥j.

total Order On elemints ust the followng 4-Increaing Onde , ', -2 5 2, 6 01 111 く21-2 14-2 -2 2 21 = 121 25 2 2-3 211-31 -ろる-5 35 oA -2 2 -3 -5 , L,-2,2,-3, -5 ) nc-cauing Ordes

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