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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
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\def\HW{4}
\def\DUE{9/27/2018}
\def\NAME{YOUR NAME HERE}
\title[Homework \HW]{Homework \HW \\
Math 308\\
\small Due: \DUE}
\author{}
%\date{}                                           % Activate to display a given date or no date

\begin{document}
%\maketitle %Comment out this line, and uncomment the following for your assignment:
\begin{flushright}
\NAME\\
Math 308\\
Homework \HW\\
\DUE
\end{flushright}


\begin{enumerate}[1.]
\item Which of the following are statements? Of the sentences that are statements, which are true and which are false? 
%Your answers should be of the form ``This is not a statement'', or ``this is a false statement''. 
\begin{enumerate}[(a)]
	\item Life is sweet.
%		\Ans{}

	\item Is $2$ a prime?
%		\Ans{}

	\item Prove that 2 is a prime.
%		\Ans{}

	\item The President of the United States in 1789 was a man.
%		\Ans{}

	\item The President of the United States in 2089 will be a woman.
%		\Ans{}

\end{enumerate}

\bigskip

\item Let $A$ be the statement ``Howard fell'' and let $B$ be the statement ``Howard broke his leg.''
\begin{enumerate}[(a)]
	\item Write English statements corresponding to each of these:
	$$A \land B, \quad \neg A \land \neg B, \quad \neg(A \land B), \quad B \land \neg A.$$
	\item For each of the statements in (a), under what circumstances would that statement be false? (Again, in sentences, using the example of Howard.)
	
%	\begin{ans}
%
%	\end{ans}
	
\end{enumerate}
\item Suppose four cards are given, each of which has a letter on one side and
a number on the other. They are displayed as follows:
\\ \smallskip
\centerline{$
	\TikZ{\draw (0,0) rectangle (1,1.5); \node at (.5,.75) {$1$};} \qquad 
	\TikZ{\draw (0,0) rectangle (1,1.5); \node at (.5,.75) {$C$};} \qquad 
	\TikZ{\draw (0,0) rectangle (1,1.5); \node at (.5,.75) {$B$};} \qquad 
	\TikZ{\draw (0,0) rectangle (1,1.5); \node at (.5,.75) {$2$};}. 
$} \smallskip
Which card(s) need not be turned over in order to determine the truth value of the statement, ``If a card has $B$ on one side, then it has 2 on the other side"?

	
%	\begin{ans}
%
%	\end{ans}
	

\item Represent the following two statements as implications.
\begin{enumerate}[(a)]
	\item  I go to the movies in the afternoon if it is rainy.
	\item I go to the movies in the afternoon only if it is rainy.
\end{enumerate}


	
%	\begin{ans}
%
%	\end{ans}
	

\item Suppose that $A$ is true and $B$ is false. Which of the following are true? 
\begin{enumerate}[(a)]
\item $A \Rightarrow B$ 
%	\Ans{}

\item $B\Rightarrow A$
%	\Ans{}

\item $\neg B \Rightarrow A$
%	\Ans{}

\item $A \Rightarrow A$ 
%	\Ans{}

\item $A \lor \neg B$
%	\Ans{}

\item $ \neg A \Rightarrow A$
%	\Ans{}

\item $\neg (A \lor B)$
%	\Ans{}

\end{enumerate}


\bigskip

\item Construct truth tables for the following, and decide whether each is a tautology, contradiction, or neither. Be sure to include all the intermediate steps when constructing your truth tables!
\begin{enumerate}[(a)]
	\item   $\neg(A \lor \neg B)$
	
	
%	\begin{ans}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% The beginnings of a truth table for your use: %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	{\def\arraystretch{1.2}
%	\[
%		\begin{array}{|c|c||c|c||c|}
%		\hline
%		A 	& B 	& \neg B 	& A \lor \neg B & \neg(A \lor \neg B) \\\hline\hline
%			&	&		& 			&		\\\hline
%		\end{array}
%	\]}

%	\end{ans}
	
	
	\item $A \Rightarrow (B \lor \neg A)$
	
%	\begin{ans}
%
%	\end{ans}
	
	\item $ B \land \neg((\neg A ) \Rightarrow B)$
	
%	\begin{ans}
%
%	\end{ans}
	
	\item $\big((\lnot A\Rightarrow B)\land (\neg A\Rightarrow \lnot B)\big)\Rightarrow A$
	
%	\begin{ans}
%
%	\end{ans}
	
\end{enumerate}


\bigskip

\item Show that each of the following are equivalent statements using truth tables. (Your truth tables will need $2^3 = 8$ rows.)
\begin{enumerate}[(a)]
	\item $A \lor (B \lor C)$ and $(A \lor B) \lor C$ (showing that $A \lor B \lor C$ is well-defined).
	
%	\begin{ans}
%
%	\end{ans}
	
	\item $A \land (B \lor C)$ and $(A \land B) \lor (A \land C)$. 
	
%	\begin{ans}
%
%	\end{ans}
	
\end{enumerate}


\bigskip

\item Negate the following statements (giving your answers in sentences).
\begin{enumerate}[(a)]
	\item $A$ is true or $B$ is false. 
%	\\
%	\Ans{}
	
	\item $A$ is false and $B$ is true.
%	\\
%	\Ans{}
	
	\item $A$ is true or $B$ is true. 
%	\\
%	\Ans{}
	
	
\end{enumerate}
\end{enumerate}





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