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\begin{center}
{
Math 2106-D, Foundations of Mathematical Proof \\
SAMPLE EXAM 1
}
\end{center}

\begin{enumerate}
\vfill
\item Let $P$ and $Q$ be statements. Fill out the truth table for $(P\implies Q)\land(Q\implies P)$. Is this a tautology, a contradiction, or neither?
\vfill
\item Show that for any sets $A,B,CD$, we have
\[
(A\times C)\cap(B\times D)=(A\cap B)\times(C\cap D)
.
\]
\vfill
\item Show that if $p$ is a prime and $a,b$ are natural numbers with $p|ab$, then $p|a$ or $p|b$. 
\vfill
\item Show that if $n$ is a composite number, then there is a prime number $p$ dividing $n$ satisfying $p\leq\sqrt n$.
\vfill
\item Show that $\sqrt[3]2$ is irrational.
\vfill
\item Show that if $x$ and $y$ are real numbers with $0<x<y$, then $\sqrt x<\sqrt y$. (Don't use calculus for this problem). 
\vfill
\end{enumerate}

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