Let, R = {(a, b): a,b ϵ Z and (a + b) is even}, then R is 

Concept:

Reflexive: Each element is related to itself.

• R is reflexive if for all x ∈ A, xRx.

Symmetric: If any one element is related to any other element, then the second element is related to the first.

• R is symmetric if for all x, y ∈ A, if xRy, then yRx.

Transitive: If any one element is related to a second and that second element is related to a third, then the first element is related to the third.

• R is transitive if for all x, y, z ∈ A, if xRy and yRz, then xRz.

R is an equivalence relation if A is nonempty and R is reflexive, symmetric and transitive.

Calculation:

Here, R = {(a, b): a,b ϵ Z and (a + b) is even}

1. Since,a + a = 2a, ,which is even, so R is reflexive.

2. If a + b is even then b + a will also be even. So, R is symmetric.

3. Let, a = 3, b = 5, and c = 7.

a + b = 3 + 5 = 8 (which is even), b + c = 5 + 7 = 12 (which is again even) and a + c = 3 + 7 = 10 (which is also even)

Therefore, R is an equivalence relation on Z.

Hence, option (2) is correct.