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Intro to Mathematical Reasoning (Math 300) -- 
Homework 7\footnote{Version 2/29/04}
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\begin{enumerate}
\item End of Chapter 4, problem 12.
\item Recall that 
a relation is a set whose elements are ordered
pairs.  Now let $A$ be a set
and let  $Q$ and $R$ be relations on $A$.
\begin{enumerate}
\item
Prove or disprove:  if $Q$ and $R$ are partial order
relations then so is $Q \cup R$.
\item
Prove or disprove:  if $Q$ and $R$ are partial order
relations then so is $Q \cap R$.
\end{enumerate}
\item
Say that two positive integers $x$ and $y$ are {\em friendly}
if there are nonzero integers $a$ and $b$ such that $a^2x=b^2y$.
Prove that friendliness is an equivalence relation on the
set of positive integers. (Note: in your proof
you are not permitted to use the fact that every integer
has a unique factorization into primes.)
\item End of Chapter 4, problem 17.
\item Prove Theorem 4.3.21
\item Exercises 5.1.10 and 5.1.12
\end{enumerate}
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