1) Prove each of the following: Assume \( n \) and \( m \) represent integers.
a) \( n^{2} \) is odd iff \( n+4 \) is odd.
b) If \( m \cdot n \) is even, then \( n \) is even or \( m \) is even.
c) Let \( x, y \in \mathbb{R} \). Prove that if x is rational and y is irrational, the \( \mathrm{x}+\mathrm{y} \) is irrational.
d) Prove or disprove the following (This means prove it if it is true and give a counterexample if it is false): if \( x \) is irrational and y is irrational, the \( \mathrm{x}+\mathrm{y} \) is irrational