Given the Equivalence relation R = { x, y $\in$ $\Bbb{Z}$ : (x+y) mod 2 = 0}, what are equivalence classes 1 and 2?
There is no general way to answer such a question other than to repeat the definition. So one might answer for example for $1$:


  The equivalence class that $1$ belongs to is the set of all integers $n$ such that $1 + n \equiv 0 \mod 2$.


You might simplify this by noting that $1 + n \equiv 0$ implies $n \equiv 2-1$ or $n \equiv 1$ modulo $2$, giving:


  The equivalence class that $1$ belongs to is the set of all integers $n$ such that $n \equiv 1 \mod 2$.




Now the answer that the asker is looking for will probably be "the odd integers". If they disagree that the above answer is correct I would argue that the question is too vague however, as in my view the above is perfectly correct.