Check the correctness of the proof if there is this proof wrong then how to prove the given theorem.   Theorem For any cardinal numbers α and β, if α<β, then there exists a nonzero cardinal number γ such that α+γ=β.

Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.

If and are positive real numbers, prove that there exist a positive integer such that . This property is called Archimedean Property of the real numbers. (Hint: If for all , then is an upper bound for the set . Use the completeness property of to arrive at a contradiction.)