Consider the following four relations defined on the set of integers Z : R₁ = (x, y) Z Z : x^2 + y^2 is an even integer R₂ = (x, y) Z Z : xy 0 R₃ = (x, y) Z Z : |x - y| 3 R₄ = (x, y) Z Z : x divides y Which of the above relations is an equivalence relation?

Step-by-step solution

An equivalence relation must be reflexive, symmetric, and transitive. For R₁ : Reflexivity: For any x Z , x^2 + x^2 = 2x^2 , which is always an even integer. Thus, x R₁ x . Symmetry: If x^2 + y^2 is even, then y^2 + x^2 is also even. Thus, x R₁ y y R₁ x . Transitivity: x^2 + y^2 is even if and only if x and y have the same parity (both are even or both are odd). If x R₁ y and y R₁ z , then x has the same parity as y , and y has the same parity as z . This implies x and z have the same parity, so x^2 + z^2 is even.