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\headers{Math303}{Homework 5}{Due: Feb 14, 2024}

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\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  Unless the problem indicates otherwise, all problems require some justification; a correct answer without supporting reasoning is not sufficient.  Submissions must be stapled.  See ``Guidelines and advice'' on the course webpage for more information.

\begin{enumerate}

\item {} [BP 2.2] Express each statement or open sentence in one of the forms $P\land Q$, $P \lor Q$, or $\sim P$.  Be sure to state explicitly what $P$ and $Q$ mean in each part.
\begin{enumerate}
	\item The matrix $A$ is not invertible.
	\item $x < y$
	\item The quiz is scheduled for Wednesday or Friday.
	\item $x\in A \cap B$
\end{enumerate}

\item {} [BP 2.3] Express each of the following as a sentence of the form ``If $P$, then $Q$'' without changing the meaning.
\begin{enumerate}
	\item For a function to be continuous, it is sufficient that it is differentiable.
	\item Whenever a surface has only one side, it is non-orientable.
	\item A geometric series with ratio $r$ converges if $|r|<1$.
	\item The discriminant is negative only if the quadratic equation has no real solutions.
	\item For a matrix to be invertible, it is necessary that no column has all zero entries.
\end{enumerate}

\item {} [BP 2.4] Express each of the following as a sentence of the form ``$P$ if and only if $Q$'' without changing the meaning.
\begin{enumerate}
	\item If a function has a constant derivative, then it is linear, and conversely.
	\item If $a\in\QQ$ then $5a\in\QQ$, and if $5a\in\QQ$ then $a\in\QQ$.
\end{enumerate}

\item {} [BP 2.5]
\begin{enumerate}
	\item Write truth tables for the following statements.
	\begin{multicols}{2}
	\begin{enumerate}
		\item $(Q\lor R) \iff (R\land Q)$
		\item $(P\land \sim P) \land Q$
	\end{enumerate}
	\end{multicols}
	\item Suppose that the statement $((P\land Q) \lor R)\implies (R\lor S)$ is false.  Determine the truth values of $P$, $Q$, $R$, and $S$.  (Hint: try to solve without using a truth table.)
\end{enumerate}

\item {} [BP 2.6]  Decide whether or not the following pairs of statements are logically equivalent.
\begin{enumerate}
	\item $(P\implies Q) \lor R$ and $\sim ((P\land \sim Q) \land \sim R)$
	\item $\sim (P\Rightarrow Q)$ and $P\land \sim Q$
	\item $P\land (Q\lor \sim Q)$ and $(\sim P) \Rightarrow (Q\land \sim Q)$
\end{enumerate}

\item Suppose that $x$ is a real number and $x=\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\ldots}}}}$.  Determine $x$.

\item You stand before a king who is completely honest and logical, but also a bit stingy.  The king is in a good mood, and promises you the following: ``If you tell me a statement which is true, then I will give you $x$ gold coins, for some integer $x$ such that $1\le x\le 500$.  If you tell me a false statement, then I will give you nothing.  If you tell me a statement which leaves me with no logically consistent options, then I will chop off your head.''  What should you say to the king to stay alive and get as many coins as possible?

\end{enumerate}

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