(1) Let r be the relation on P defined by ary if and only if x divides y. Let s be the relation on P defined by xsy if and only if a and y are both even or both odd. (a List 5 elements of the relation r. (b) List 5 elements of the relation s. (c) List 3 elements in r n s. (2) Let r = {(a, b) E Z x Z | a + b is even}. If x E Z is odd, then what is the set ty E Z | (x, y) Er}? Justify your answer with a short reason written as a complete sentence. (3) Let r be the relation on Z defined by cry if and only if 3 divides x - y. (a) List 5 elements of the relation r. (b) Explain why (x, xx) Er for every x E Z. (c) If b E Z and (0, b) Er, prove that be {3n n E Z}.3) Let r be the relation on Z defined by xcry if and only if y = 2x. (a) Is r reflexive? If so, explain why. If not, show why not. (b) Is r symmetric? If so, explain why. If not, show why not. (c) Is r antisymmetric? If so, explain why. If not, show why not. (d) Is r transitive? If so, explain why. If not, show why not. (4) Let r be the relation on Z defined by ary if and only if x + y is odd. (a) Is r reflexive? If so, explain why. If not, show why not. (b) Is r symmetric? If so, explain why. If not, show why not. (c) Is r antisymmetric? If so, explain why. If not, show why not. 3 (d) Is r transitive? If so, explain why. If not, show why not.(1) Let X = {1,2, 3}. Define the relation r on the power set P(X ) by: Ar B if and only if A C B. List 6 of the elements of r. (2) Let r be the relation on P defined by arb if and only if god(a, b) = 1. (a) Is r reflexive? If so, explain why (using a com- plete sentence and correct terminology / notation). If not, show why not by finding an element x E Z so that (x, x) Er. (b) Is r symmetric? If so, explain why (using a complete sentence and correct terminology / notation). If not, show why not by finding an element (x, y) Er so that (y, x) Er. (c) Is r antisymmetric? If so, explain why (us- ing a complete sentence and correct terminol- ogy / notation). If not, show why not by finding an element (x, y) Er so that (y, x) Er and x ty. (d) Is r transitive? If so, explain why (using a com- plete sentence and correct terminology / notation). If not, show why not by finding an element (x, y) Er and an element (y, z) Er so that (x , z) er