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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
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\def\HW{5}
\def\DUE{10/2/2018}
\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 The connective ``unless'' can be ambiguous, and this exercise will pinpoint the ambiguity.

\smallskip

\emph{We awake at dawn, and we are told}

\centerline{``$\underbrace{\text{\emph{We will have a picnic today}}}_{A}$ \quad unless \quad 
$\underbrace{\text{\emph{it is raining at 10 A.M.}}}_{B}$''}


\medskip

\noindent Let $A \star B$ denote $A$ unless $B$ (not a standard notation). Complete as much of a truth table as possible for$A \star B$, and discuss any ambiguous lines.



%\begin{ans}

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

%\end{ans}

\item Use a truth tables to check that the following are tautologies.
\begin{enumerate}[(a)]
\item $\big( (A \Rightarrow B) \land (B \Rightarrow C) \big) \Rightarrow (A \Rightarrow C)$.

%\begin{ans}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% The beginnings of a truth table for your use: %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%{\def\arraystretch{1.2}
%\[
%	\begin{array}{|c|c|c||c|c|c||c||c|}
%	\hline
%	A 	& B 	& C & A \Rightarrow B & B \Rightarrow C 
%		& \begin{array}{l}(A \Rightarrow B) \\ 
%					\quad \land (B \Rightarrow C)\end{array}  
%		& A \Rightarrow C 
%		& \begin{array}{r}\big((A \Rightarrow B) \land (B \Rightarrow C)\big)\\
%					 \Rightarrow (A \Rightarrow C) \end{array} 
%					\\\hline\hline
%	T & T & T &
%			&	&	& 	& 	\\\hline
%	T & T & F &
%			&	&	& 	& 	\\\hline
%	T & F & T &
%			&	&	& 	& 	\\\hline
%	T & F & F &
%			&	&	& 	& 	\\\hline
%	F & T & T &
%			&	&	& 	& 	\\\hline
%	F & T & F &
%			&	&	& 	& 	\\\hline
%	F & F & T &
%			&	&	& 	& 	\\\hline
%	F & F & F &
%			&	&	& 	& 	\\\hline
%	\end{array}
%\]}
%
%\end{ans}




\item  $( A \Rightarrow B ) \Leftrightarrow ( \neg B \Rightarrow \neg A )$.

%\begin{ans}

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

%\end{ans}


\end{enumerate}


\item Rewrite the following as ``if\dots, then\dots'' statements.
\begin{enumerate}[(a)]
\item A sufficient condition for Peter to win the Championship is that he wins in Brazil. 

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

\item To be President of the US it is necessary to be born in the US.

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

\item To be Prime Minister of India it is not necessary to be born in India.

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

\item You can't make an omelette without breaking some eggs.

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

\item A masters degree is required for this position.

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

\item To cross this river, we just need to find a boat. (Careful!)

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


\end{enumerate}

\item Let $A$ be the statement ``$x^2- 2x - 3 >0$" and $B$  be the statement ``$x > 3$''. Which of the following are true, which are false? Be sure to justify your overall answer. 


\begin{center}
$A \Rightarrow B, \qquad B \Rightarrow A, \qquad 
	\neg A \Rightarrow \neg B, \qquad \neg B \Rightarrow \neg A,$

\smallskip

``$A$ is necessary for $B$'', \qquad ``$B$ is necessary for $A$'', 

\smallskip

``$A$ is sufficient for $B$'', \quad and \quad ``$B$ is sufficient for $A$''. 

\smallskip
\end{center}


%\begin{ans}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% A graph of y = x^2 - 2x - 3 for your use: %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%$$\begin{matrix}
%\text{Graph of } y = x^2 - 2x - 3:\\~\\
%\begin{tikzpicture}
%\begin{axis}[
%        axis y line=center,
%        axis x line=middle, 
%        axis on top=true,
%        xmin=-2.5,
%        xmax=4.5,
%        ymin=-5.5,
%        ymax=9.5,
%        height=1.5in,
%        width=3in,
%        xtick={-2,...,4},
%        ytick={-5,0,...,20},
%    ]
%    \addplot [domain=-5:5, samples=50, mark=none, ultra thick, ] {(x)^2- 2*(x)-3};
%\end{axis}
%\end{tikzpicture}\end{matrix}$$

%\end{ans}


\item Which of the following statements are equivalent? For each, also give write the sentence in terms of logical notation, with $A$ being the statement ``my team won the last game'' and $B$ the statement ``my team won the championship''. (Assume ties are not an option, so that ``didn't win'' is the same thing as ``lost''.) 

\begin{enumerate}
	\item If my team lost the last game, then they must have lost the championship. 
	%\Ans{}
	\item If my team lost the last game, then your team won the championship.
	%\Ans{}
	\item If my team lost the last game, then they won  the championship.
	%\Ans{}
	\item If my team won the championship, then they won the last game.
	%\Ans{}
	\item If my team won the last game, then they won the championship.
	%\Ans{}
	\item If my team lost the championship, then they must have lost the last game.
	%\Ans{}
\end{enumerate}

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


\item What are the negation, inverse, converse, and contrapositive  of each of the following? 

\begin{enumerate}
	\item If $x > 5$, then $X$ is red.

%	\begin{ans}
%	\begin{enumerate}[\hspace{.75in}]
%		\item[\bf Negation:] 
%		\item[\bf Inverse:] 
%		\item[\bf Converse:] 
%		\item[\bf Contrapositive:] 
%	\end{enumerate}
%	\end{ans}

	\item Taking a shower is necessary for me to be happy all day.

%	\begin{ans}
%	\begin{enumerate}[\hspace{.75in}]
%		\item[\bf Negation:] 
%		\item[\bf Inverse:] 
%		\item[\bf Converse:] 
%		\item[\bf Contrapositive:] 
%	\end{enumerate}
%	\end{ans}

	\item It isn't necessary to understand things to argue about them.

%	\begin{ans}
%	\begin{enumerate}[\hspace{.75in}]
%		\item[\bf Negation:] 
%		\item[\bf Inverse:] 
%		\item[\bf Converse:] 
%		\item[\bf Contrapositive:] 
%	\end{enumerate}
%	\end{ans}

	\item Stop doing that, or I'll get angry. 

%	\begin{ans}
%	\begin{enumerate}[\hspace{.75in}]
%		\item[\bf Negation:] 
%		\item[\bf Inverse:] 
%		\item[\bf Converse:] 
%		\item[\bf Contrapositive:] 
%	\end{enumerate}
%	\end{ans}

	\item  $A \Rightarrow (B \Rightarrow C)$ (Simplify your answer.)

%	\begin{ans}
%	\begin{enumerate}[\hspace{.75in}]
%		\item[\bf Negation:] 
%		\item[\bf Inverse:] 
%		\item[\bf Converse:] 
%		\item[\bf Contrapositive:] 
%	\end{enumerate}
%	\end{ans}

\end{enumerate} 




\item Let $a$ and $b$ be integers. Prove the following.\\
{[\emph{You may take for granted that if $a$ is even, then $a = 2k$ for some integer $k$; that if $a$ is odd then $a = 2\ell + 1$ for some integer $\ell$; and that no number can be both even and odd.}]}
\begin{enumerate}
\item We have $a$ is odd if and only if $a+2$ is odd.

%	\begin{proof}
%	\end{proof}

\item We have $a$ is odd if and only if $a^3$ is odd.

%	\begin{proof}
%	\end{proof}

\item We have $ab$ is odd if and only if $a$ and $b$ are both odd.


%	\begin{proof}
%	\end{proof}


\end{enumerate}


\item Rewrite the following using $\forall$ and $\exists$.
\begin{enumerate}[(a)]
\item For all integers $x$, we have $x$ is odd or even. \\
(Your answer should include a definition of even/odd using $\exists$.)

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


\item There exist two positive numbers such that their sum is negative.

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

\end{enumerate}





\item Consider the statement ``If $a$ and $b$ are real numbers with $a \neq 0$, then $ax+b=0$ has a solution.'' 
\begin{enumerate}[(a)]
\item Rewrite this statement  using symbolic notation $\forall$ and $\exists$.

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

\item Negate this statement, giving your answer both in symbolic notation, and in words. 

%\begin{ans}
%\begin{enumerate}[\qquad\qquad\qquad]
%\item[\bf In symbols:]
%\item[\bf In words:]
%\end{enumerate}
%\end{ans}

\end{enumerate}

\item Negate the following.
\begin{enumerate}[(a)]
\item There exists a grey cat.

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

\item Every cat has an owner. 

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

\item Some of the students in the class are not here today.

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

\item For all $x, y \in \ZZ_{>0}$ there exists $z \in \ZZ_{>0}$ such that $x=y+z$.

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

\item The number $\sqrt{x}$ is rational if $x$ is an integer.

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

\end{enumerate}

\item For each of the following,
\begin{enumerate}[\qquad\qquad(i)]
\item restate in words;
\item decide whether it's true or false; and 
\item prove or disprove accordingly.
\end{enumerate}

\begin{enumerate}[(a)]
\item $\forall x \in \ZZ, \exists y \in \ZZ (x^2 = y)$

%\begin{ans}
%\begin{enumerate}[(i)]
%\item \textbf{In words:} 
%\item This statement is \fbox{true/false}. 
%\item \begin{proof}
%
%
%\end{proof}
%\end{enumerate}
%\end{ans}

\item $\forall y \in \ZZ, \exists x \in \ZZ (x^2 = y)$

%\begin{ans}
%\begin{enumerate}[(i)]
%\item \textbf{In words:} 
%\item This statement is \fbox{true/false}. 
%\item \begin{proof}
%
%
%\end{proof}
%\end{enumerate}
%\end{ans}

\item $\exists x \in \ZZ, \forall y \in \ZZ (x^2 = y)$

%\begin{ans}
%\begin{enumerate}[(i)]
%\item \textbf{In words:} 
%\item This statement is \fbox{true/false}. 
%\item \begin{proof}
%
%
%\end{proof}
%\end{enumerate}
%\end{ans}

\item $\exists y \in \ZZ, \forall x \in \ZZ (x^2 = y)$

%\begin{ans}
%\begin{enumerate}[(i)]
%\item \textbf{In words:} 
%\item This statement is \fbox{true/false}. 
%\item \begin{proof}
%
%
%\end{proof}
%\end{enumerate}
%\end{ans}

\end{enumerate}
\end{enumerate}



\end{document}