A relation R is defined on the set of integers as xRy if (x + y) is even.…

CONCEPT — An equivalence relation is reflexive, symmetric, and transitive. A congruence relation partitions a set into classes whose members share the same residue under the chosen modulus.

Application

1. Rewrite the condition modulo 2: x + y is even exactly when x and y have the same parity, equivalently x ≡ y (mod 2).

2. Reflexivity: for every integer x, x + x = 2x is even, so xRx.

3. Symmetry: if xRy, then x + y is even. Since y + x = x + y, yRx.

4. Transitivity: if xRy and yRz, then x and y have the same parity and y and z have the same parity. Therefore x and z have the same parity, so xRz.

5. The class represented by 0 contains the even integers, while the class represented by 1 contains the odd integers. Every integer belongs to exactly one of these classes.

Cross-check

The remainder of an integer on division by 2 is either 0 or 1. Two integers are related exactly when these remainders agree, so the relation partitions all integers without overlap.

Result

Thus R is an equivalence relation with exactly 2 equivalence classes: the even integers and the odd integers.