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\textbf{\Large Math 3345 - Winter 2020 - Taylor\\ Assignment \asstnumber}\\
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\textbf{\Large Math 3345 - Autumn 2016 - Katz\\ Assignment \asstnumber\\ Solutions}
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\begin{document}

\toptop

\noindent
\begin{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% From Katz assignment 2
\item Let $P$ and $Q$ be sentences. Show that $\neg(P \Rightarrow Q)$ is logically equivalent to $P\wedge \neg Q$ by means of a truth table.

\item Without changing their meanings, convert each of the following (true) sentences into a statement of the form ``If P, then Q'':
\begin{enumerate}
\item An integer is divisible by $8$ only if it is divisible by $4$.
\item A geometric series with ratio $r$ converges if $|r|<1$.
\item For a function to be continuous, it is sufficient that it is differentiable.
\item For a series to be convergent, it is necessary that its terms go to zero.
\end{enumerate}

\item Explain why the sentence ``If $0=1$, then Prof. Taylor is a CIA agent'' is true using the definition of implication.

\item Write the contrapositive and the converse of each of the following statements:
\begin{enumerate}
\item If it is raining, then the ground is wet.
\item If $a=5$ then $a^2=25$
\item If $a\neq b$, then $a^4\neq b^4$.
\end{enumerate}
Do not worry about the truth or falsity of these statements.

%\item Work out a truth table for the sentence ``$(P\wedge \neg Q)\Rightarrow R.$''

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% From Katz assignment 3
%\item Write the following as if-then statements
%\begin{enumerate}
%\item The sum of an odd integer and an even integer is odd.
%\item If you add one to an even integer, you get an odd integer.
%\end{enumerate}

\item Prove that if $m$ and $n$ are odd integers, then $m+n$ is even.

\item Prove the following (you can use notes from class):
\begin{quote}
Let $m$ be an integer. If $m^2$ is even, then $m$ is even.
\end{quote}

\item Let $m$ be an integer. By proving the contrapositive, show the following statement is true:
\begin{quote}
If $m^2$ is odd, then $m$ is odd.
\end{quote}

\item Consider the statement
\begin{quote}
Let $n$ be an integer. If $n^2$ is divisible by $4$ then $n$ is divisible by $4$.
\end{quote}
Show that this statement is false by finding an integer $n$ such that $n^2$ is divisible by $4$ but $n$ is not divisible by $4$. Explain why this shows that the above statement cannot be true in terms of the negation of a conditional.

\item An integer $n$ is said to be a {\em square} if there exists an integer $m$ such that $n=m^2$. Consider the statement, ``the product of two squares is a square.'' This sentence can be written as an if-them statement as:
\begin{quote}
Let $n_1$ and $n_2$ be integers. If $n_1$ and $n_2$ are squares then $n_1n_2$ is a square.
\end{quote}

\begin{enumerate}
\item Prove the if-then statement. Hint: It's useful to write $n_1=m_1^2, n_2=m_2^2$ for integers $m_1,m_2$

\item \label{i:converse} Write the converse of the if-then statement.

\item Is the converse you write in (\ref{i:converse}) true? If yes, give a proof. If no, explain why.
\end{enumerate}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% From Katz assignment 4

%\item For each of the following sentences, state whether it is true or false, and justify your statement.
%\begin{enumerate}
%\item $(\exists x\in\R)(2x+7=3)$.
%\item $(\forall x\in \R)(2x+7=3)$.
%\item $(\exists x>0)(2x+7=3)$.
%\item $(\forall x>0)(2x+7=3)$.
%\item $(\exists x\in\R)(x^2-4x+3>0)$.
%\item $(\forall x\in\R)(x^2-4x+3>0)$.
%\item $(\exists x\geq 7)(x^2-4x+3>0)$.
%\item $(\forall x\geq 7)(x^2-4x+3>0)$.
%\end{enumerate}

%\item For each of the following statements:
%\begin{enumerate}
%\item $(\forall x\in \Z)(x \text{ is even})$
%\item $(\exists x\in \Z)(x^3>0)$
%\item $(\exists x\in \R)(x^2+1=0)$
%\item $(\forall x\in \R)(x^2+1>0)$
%\end{enumerate}
%do the following:
%\begin{enumerate}
%\item Write the statement as an English sentence.
%\item Is the statement true or false? Why?
%\item Write the negation of the statement as an English sentence.
%\item Write the negation of the statement in symbols.
%\end{enumerate}

\item By finding a counterexample, show why the following statement is false:
For each integer $n$, $(n^2+n+1)$ is a prime number.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% From Katz assignment 5
%\item Which if any of the following are true?
%\begin{enumerate}
%\item $\left((\exists x\in\R)(2x+7=15)\right)\wedge \left((\exists x\in\R)(x^2=4)\right)$.
%\item $(\exists x\in\R)\left((2x+7=15)\wedge (x^2=4)\right)$.
%\end{enumerate}
%
%\item Which if any of the following are true?
%\begin{enumerate}
%\item $\left((\exists x\in\R)(2x+7=15)\right)\wedge \left((\exists x\in\R)(x^2=16)\right)$.
%\item $(\exists x\in\R)\left((2x+7=15)\wedge (x^2=16)\right)$.
%\end{enumerate}
%
%\item For each of the following sentences, write out what it means in words, state whether it is true or false, and prove your statement.
%\begin{enumerate}
%\item $(\exists y\in\R)(\forall x\in \R)(x+y=x)$
%\item $(\forall x\in \R)(\exists y\in\R)(x+y=x)$
%\item $(\exists y\in \R)(\forall x\in \R)(x+y=0)$
%\item $(\forall x\in \R)(\exists y\in \R)(x+y=0)$
%\end{enumerate}
%
%
%\item Write the negation of the sentence
%\[(\forall x\in \R)(\exists y\in \R)(xy=1).\]
%Which is true, the original sentence or its negation?
%
%\item Are the following sentence true or false? Justify your answer.
%\begin{enumerate}
%\item  $(\exists m\in\Z)(\exists n\in \Z)(mn=20)$.
%\item $(\exists m\in \Z)(\exists n\in \Z)\left((m\neq 1)\wedge (n\neq 1) \wedge (m\neq -1)\wedge (n\neq -1)\wedge (mn=7)\right)$.
%\end{enumerate}
%





\end{enumerate}









\end{document}
