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)

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)

Given:(1) Let R1 and R2 be two transitive relations on a set A. Then, R1 UNION R2 is a transitive relation on A.(2) Let R be the relation x + y = 0 on the set of all real numbers. Then, R is antisymmetric.Select one of the following choices:Question 10Answera.(1) is True and (2) is Falseb.(1) and (2) are Truec.(1) and (2) are Falsed.(1) is False and (2) is True

Let A = {1, {1}, 3, 4, {5, 6}} be a set, and let B and C be the following relations on A:            B = {(1, 3), (4, 1), (3, {5, 6}), ({1}, 4), ({5, 6}, {1})}            C = {(1, 1), (3, 4), (4, 3), ({5, 6}, {1})The relation B is antisymmetric.a.Trueb.False

rue or false (give reasons):(a) For a symmetric relation R on a nonempty set X, xRy and yRx for each x, y ∈ X(b) For an antisymmetric relation R on a nonempty set X:i. xRy and yRx for each x, y ∈ Xii. R is also a symmetric relation

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 symmetric. Which one of the following is a valid proof showing that R is symmetric?a.Let x, y ∈ Z≥ be given.Suppose (x, y) ∈ R then x2 + y2 = 2k for some k ≥0.ie y2 + x2 = 2k for some k ≥0.thus (x, y) ∈ R.b.Let x, y ∈ Z≥ be given.Suppose (x, y) ∈ R then x2 + y2 = 2k for some k ≥0.ie y2 + x2  = 2k for some k ≥0.thus (y, x) ∈ R.c.Let x, y ∈ Z≥ be given.Suppose (x, y) ∈ R then x2 + y2 = 2k for some k ≥0.thus (y, x) ∈ R.d.Let x, y ∈ Z≥ be given.Suppose (x, x) ∈ R then x2 + y2 = 2k for some k ≥0.ie y2 + x2 = 2k for some k ≥0.thus (y, y) ∈ R.