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\title{CS243 Homework Assignment 3}
\date{Due Tuesday, September 25}


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 Please hand in a hard copy of your solutions before class on the due date. The
answers to the homework assignment should be typeset using LaTeX and must be your own individual work. You may
discuss problems with  other students in the class; however, your write-up must
mention the names of these individuals. 
 
\begin{enumerate} 
\item (40 points, 8 points each) Prove or disprove each of the following
statements below. If you believe the statement is true, prove the statement
using one of the proof techniques we
discussed in class and clearly state the proof strategy you use. If you believe
the statement 
if false, disprove the claim by proving a counterexample.

\begin{itemize}
\item The product of an odd and even integer is even. 
\item If $r$ is a rational number and $p$ is an irrational number, then $r+p$ is
irrational.
\item If $r$ is a rational number and $p$ is an irrational number, then $rp$ is
irrational.
\item If $7x+9$ is even, then $x$ is odd.
\item For all real numbers $a,b,c$, ${\rm max}(a, {\rm max}(b,c)) = {\rm
max}({\rm max}(a, b),c)$
\end{itemize}
\item (5 points) What is the Cartesian product of the sets $\{ \emptyset \}$ and $\{ a,b \}$
\item (5 points) Prove or disprove: There exists a set $S$ whose powerset is equal to $\{ S \}$
\item (15 points) Prove that $A \cup (A \cap B) = A$. Your proof should {\bf
not} use any of the set identities from Table 1, p. 124 of the Rosen book.
\item (15 points) Prove or disprove: The function $f$ defined by $f(x) = 2x+5$ is a bijection 
from $\mathbb{R}$ to $\mathbb{R}$.
\item (5 points) Prove or disprove: The function $f$ defined by $f(x) = 2x+5$ is a bijection 
from $\mathbb{Z}$ to $\mathbb{Z}$.
\item (15 points) Let $x$ be a real number. Prove the following:
 \[ \lfloor 3x \rfloor = \lfloor x \rfloor + \lfloor x + \frac{1}{3} \rfloor + 
\lfloor x + \frac{2}{3} \rfloor \]
\end{enumerate}
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