91Ó°ÊÓ

Prove: \(x=y\) if and only if \(x y=\frac{(x+y)^{2}}{4}\). Note, you will need to prove two "directions" here: the "if" and the "only if" part.

Prove that there are no integer solutions to the equation \(x^{2}=4 y+3\).

. Prove that every prime number greater than 3 is either one more or one less than a multiple of 6 .

Simplify the statements below to the point that negation symbols occur only directly next to predicates. (a) \(\neg \forall x \forall y(x

Suppose \(P\) and \(Q\) are (possibly molecular) propositional statements. Prove that \(P\) and \(Q\) are logically equivalent if any only if \(P \leftrightarrow Q\) is a tautology.

Suppose \(P_{1}, P_{2}, \ldots, P_{n}\) and \(Q\) are (possibly molecular) propositional statements. Suppose further that $$\begin{array}{cl} & P_{1} \\\& P_{2} \\\& \vdots \\\& P_{n} \\\\\hline \therefore & Q\end{array}$$ is a valid deduction rule. Prove that the statement $$\left(P_{1} \wedge P_{2} \wedge \cdots \wedge P_{n}\right) \rightarrow Q$$ is a tautology.