Let R be the relation on Z≥ (the set of integers) defined by (x, y) ∈ R iff x2 + y2 = 2k for some integers k ≥0.Which one of the following is an ordered pair in R?a.(1, 0)b.(2, 9)c.(3, 8)d.(5, 7)

Let R be the relation on Z≥ (the set of integers) defined by (x, y) ∈ R iff x2 + y2 = 2k for some integers k ≥0.R is not antisymmetric.Which of the following ordered pairs can be used together in a counterexample to prove that R is not antisymmetric? (Remember that R is defined on Z≥)a.(–1, 1) and (1, –1)b.(5, 9) and (13, 15)c.(8, 7) and (7, 8)d.(3, 1) and (1, 3)

A = {0, 1, 2, 3, 4, 5, 6, 7} ,suppose R and T are Two relations on A such that R = {(x, y) : 2 * x + 3y = 15} ,T= {(x,y):3x+2y in A} Write down R,T and R°T as a set of ordered pairs?

Q1. Consider the relation R on the set of integers as xRy if and only if x<y. Then prove that R is partial order relation.

etermine whether the each of the relation defined on the set of positive integers is reflexive,symmetric, antisymmetric, or transitive.(a) R = {(x, y) : xy = 2}(b) R = {(x, y) : xy ≥ 1}(c) R = {(x, y) : x = and2}(d) R = {(x, y) : 3 divides (x + 2and)}(It is) R = {(x, y) : x − and = 2}(f) R = {(x, y) : 3 divides (x − an

Let R be a relation on the integers, where R=