- a) Use proof by contradiction to prove that there are no positive integers, \(x\) and \(y\), such that
$$x ^ { 2 } - y ^ { 2 } = 1$$
b) Prove, by counter-example, that the statement
\section*{" if \(a\) is rational and \(b\) is irrational then \(\log _ { a } b\) is irrational"}
is false.
c) Use algebra to prove by exhaustion that for all \(n \in \mathbb { N }\)
$$\text { " } n ^ { 2 } - 2 \text { is not a multiple of } 4 "$$
\section*{(Total for Question 10 is 6 marks)}