Question

QI. Let A-(-4-3-2-1,0,1,2,3,4]. R İs defined on A as follows: For all (m, n) E A, mRn㈠4](rn2_n2) Show that the relation R is an equivalence relation on the set A by drawing the graph of relation Find the distinct equivalence classes of R. Q2. Find examples of relations with the following properties a) Reflexive, but not symmetric and not transitive. b) Symmetric, but not reflexive and not transitive. c) Transitive, but not reflexive and not symmetric. d) Reflexive and symmetric, but not transitive. e) Reflexive and transitive, but not symmetric f) Symmetric and transitive, but not reflexive.

Answer 1

Given: A = { - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4} R is defined on A as follows (m, n) elementof A, m R n Doubleheadarrow 4/m^2 - n^2 Here every line is both way directed. Since every element has self loop, the relation is a reflexive relation. Since every line is both way directed, the relation is a symmetric relation. \r\n From the graph it is clear that the relation is a transitive relation. Therefore The given relation on A is an equivalence relation. The distinct equivalence classes of R are { - 4, - 2, 0, 2, 4} and { - 3, - 1, 1, 3}