Question

1. Define a relation on Z by aRb provided a -b a. Prove that this relation is an equivalence relation. b. Describe the equivalence classes. 2. Define a relation on Z by akb provided ab is even. Use counterexamples to show that the reflexive and transitive properties are not satisfied 3. Explain why the relation R on the set S-23,4 defined by R - 11.1),(22),3,3),4.4),2,3),(32),(2.4),(4,2)) is not an equivalence relation.

Answer 1

"Define a relation on z by a^R b provided a^2 = b^2 (a) I) To prove Ris reflexive. Let a element of Z then a^2 = b^2 Therefore A^R a Therefore R is reflexive. Ii) To prvoe R is symmetric Let a,b element of z. a^2 = b^2 Implies a^Rb Implies b^2 = a^2 Implies bRa Therefore R is symmetric. Iii) To prove R is transitive. Let a,b,c elementofz s.t a^Rb + b^Rc I.e a^2 = b^2 & b^2 = c^2 Implies a^2 = c^2 Implies a R c Therefore R is Transitive Therefore from (i)(ii) &(iii) R is an equivalence relation (b) Now equivalence classes [x] = {Y element of zs.t c^2 = y^2} [x] = {y elementof z s.t y = plusminus x} aRb provided ab is even Let 1 elementof z then 1.1 = 1 is not even therefore