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%\pagestyle{fancyplain} \chead{Math 3345, Autumn 2021, Kutler}


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\begin{center}
{\large{\textsc{Homework 3}}} \\
\textsc{Math 3345 -- Autumn 2022 -- Kutler}
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Please complete the following problems on your own paper. Solutions should be written clearly, legibly, and with appropriate style.

\begin{enumerate}
\item Write \textbf{both} the contrapositive and the converse of each conditional sentence below.
\begin{enumerate}
\item If it is raining, then the ground is wet.
\item If $a = 4$, then $a^2 = 16$.
\item If $a \neq b$, then $a^4 \neq b^4$.
\end{enumerate}
Do not worry about the truth value of any of these statements.

\item \textbf{{[Falkner Section 2 Exercise 7]}} Let $x$ and $y$ be real numbers.
\begin{enumerate}
\item Let $A$ be the sentence ``If $x + y > 0$, then $x > 0$ or $y > 0$.'' Use Theorem 2.10 and one of De Morgan's laws to show that $\neg A$ is logically equivalent to ``$x + y > 0$ and $x \leq 0$ and $y \leq 0$.'' Be careful not to skip any steps.

\item Is the sentence $A$ in part (a) true, or is $\neg A$ true? Explain why.

\item Let $B$ be the sentence ``If $x + y > 2$, then $x > 2$ or $y > 2$.'' Is $B$ true, or is $\neg B$ true, or is it impossible to say without further information about the specific values of $x$ and $y$? (Hint: Can you find specific values for $x$ and $y$ for which $B$ is true? If so, give an example of such values. Can you find other specific values for $x$ and $y$ for which $\neg B$ is true? If so, give an example of such values.)
\end{enumerate}

\end{enumerate}

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\begin{center}
\textbf{Practice Problems}
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It is strongly recommended that you complete the following problems. There is no need to write up polished, final versions of your solutions (although you may find this a useful exercise). Please do not submit any work for these problems.

\begin{enumerate}
\item Use De Morgan's laws to find a sentence which is logically equivalent to
\[ \neg \Big[ \big( (x > 1) \wedge (x < 3) \big) \vee \big( (x \geq 4) \wedge (x < 7) \big) \Big]\]
and which does not use the logical connective ``$\neg$''.

\item Is $(P \Rightarrow Q) \Rightarrow R$ logically equivalent to $P \Rightarrow (Q \Rightarrow R)$? Use a truth table to justify your answer.
\end{enumerate}




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