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\begin{center}
{\large{\textsc{Homework 1}}} \\
\textsc{Math 3345 -- Spring 2024 -- Kutler}
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\begin{center}
\textbf{Exercises}
\end{center}

Please complete the following problems on your own paper. Solutions should be written clearly, legibly, and with appropriate style.

\begin{enumerate}
\item \textbf{{[Falkner Section 2 Exercise 1]}} Let $P$ and $Q$ be logical sentences. Prove the second DeMorgan Law
\[ \neg (P \vee Q) \ \text{is logically equivalent to} \ \neg P \wedge \neg Q \]
in two ways:
\begin{enumerate}
\item By means of a truth table;
\item By means of an explanation in words.
\end{enumerate}

\item Let $P$, $Q$, and $R$ be logical sentences. Show by means of a truth table that the sentence $P \wedge (Q \vee R)$ is not logically equivalent to the sentence $(P \wedge Q) \vee R$.

\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 Explain why the sentence
\begin{displayquote}
``This sentence is false.''
\end{displayquote}
is not a logical sentence.

\item For each sentence below, describe (in interval notation) the set of real numbers $x$ for which the sentence is true.
\begin{enumerate}
\item $(x>1) \wedge(x<3)$
\item $(x>1)\vee (x<3)$
\item $\neg((x>1)\wedge (x<3))$
\item $\big(\neg\left((x>1)\wedge (x<3)\right)\big)\vee \left((x<2)\wedge(x>1)\right)$.
\end{enumerate}
For example, the sentence $(x\geq 0)\wedge (x<5)$ is true for exactly the real numbers $x$ in the half-open interval $[0,5)$.

You may have to write your answer as a union of intervals.
\end{enumerate}




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