Question

and are both relations defined over X , and we also know that RAS is reflexive, then what do we know for sure about

Answer 1

SOLUTION:

Solution reflexive, then kns is reflexure. 1. If Rs are both proof! Suppose QeA since Rus are both reflexive on A (A, WER and (@,aES since ca,ad es in both R and so ca,aderns, so ros is reflexive- 2. If Ris are both irreflexive, then Rises irrobledive proof! suppose QCA, since Ris are both irrefledure on A, (a,0 R and @ajs - since ca,a) es in neither R nors, ca,ad & Rns , so Rns is erreflexive - 3. It R, S are both symmelrec, then ras is symmetroc - proofs suppose CQ, b) EROS Becacuo Cable ROS,(a,b) ER and CAD JE S Since Ris are both symmetric (ba) ER and Cb,aes. Since Cb, a) er and 16,QJES, (bas ERNS, SOROS is symmetric 4. lof Ros are both antisymmetric Ros is antisymmetric proof: Suppose ca,b) EROS and CbQIEROS. We need to show that arb.: Becacue (a,b) EROS and (b,a] ERNS, We know and cableR and Cb, ajer, since R es antisymmetric, az b. We have shocon that (a, b) Erns, (b, adfRAS, SO RAS is asy Cb, a) EROS, Qzb, SO RAS is antisymmetric 5. If Res are both transitive then Ros ºs transitives Proofs suppose (a,b)EROS and (b)CEROS. we need to show cacerns Because Cable ROS, we know and ca, b2eR and Ca; b) Es. Because CbC2 EROS, We know and CbCDER and (b, c) ES spoce (a,b) ER and CCER and R 85 transitive we know that (AZCIER. since cables and (b)ces and sis transitive we know th at (CIES- since Casc) ER and (a,c) Es, (a, C) EROS :-. Caberns and Cb, CIERNS, (QC) EROS, SO ROS is transitive

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