Question

*ESPECIALLY PART D PLEASE

111111 1. Let R be a relation on RxR defined by (a,b)R(c,d) if and only if a - b = c-d DIDUD a) (5 points) Prove that is an equivalence relation on RxR. b) (5 points) Describe all ordered pairs in the equivalence class of (0,0) c) (5 points) Describe all ordered pairs in the equivalence class of (3,1) d) (5 points) Describe the partition of Rx Rassociated with R.

Answer 1

Thank you. I explained part d .if u have any doubts pls comment..

(d) partition of RxR associated with R. A relation on a set Aps an equivalence relation it it is deflexive, symmetric, and transitive. We often use the tidle notation and to denote a gelation Hence, two positive real numbers are related it and only if they have the same decimal parts. It is easy to verify that 'N' PS an equivalence relation, and each equivalence class [x] consists of all the positive real numbers having the same decimal parts as a. hay. which means that the equivalence classes [17, where xe (01], form a Partition of R. ( 0 6 0 Det: Let (a,b) ER then RXR can be divided into equivalence clasy such that larbl=k ; Here k=0,1,2...n Helle partition of RXR associated with R farbl=2 la-blas label la-slo b al on RXR relation an equivalence for 'R' is Reflexive: Let coib) ERXR then lo-bl = lo-bl therefore (a,b) R (aib) Hence Ries a replexive Relation symmetric : Let Calb) & (cid) then land=1 cod #ledla la bla (ad) a caib atalne Hence 'R' is a symmetric relation Transitive: Let (a,b) R (ad) or (ad) Raib) = 10-d=le-d) & Icadl = let) loobl = le-t ; (915) R (e,t) Hence 'R' es a transitive Therefore 'R' PS a equivdance Relation. Relation