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\title{Math 214 Homework 2}
\date{Due:  noon Sept 14, 2012}
\author{Your name}


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\maketitle

Solve the following problems. Please remember to use complete sentences and good grammar. 

\begin{enumerate}
\item (4 points) In each of the following, two open sentences $P(x)$ and $Q(x)$ over a domain $S$ are given.  For each part, determine $T = \{x \in S : P(x)\Rightarrow Q(x)\}$.
 \begin{enumerate}
 \item $P(x):x-3=4;  Q(x): x\ge 8;  S=\R$. 
 \item $P(x): x\in [-1,2];  Q(x): x^2\le 2;  S=[-1,1]$.  
 \end{enumerate}


\item (4 points) Let $P, Q$ be statements. Show that $\sim(P\implies Q)$ and $P\wedge (\sim Q)$ are logically equivalent using truth table. 

\item (6 points) Write the statements so that there are no $\sim$ symbols. Then, rewrite the statements so that there are no $\forall, \exists, \in$ or $=$ symbols. 
\begin{enumerate}
\item $\sim(\forall x\in \R, \exists y\in \R, xy=1)$;
\item $\sim(\exists y\in \R, \forall x\in \R, xy=0)$;
\item $\sim(\exists n\in \Z, \exists m\in \Z, m\le n)$;
\end{enumerate}

\item (2 points) Consider the sentence, ``For every integer $n>0$ there is some real number $x>0$ such that $x<1/n$.'' Without using words of negation, write a complete sentence that negates the sentence. Which sentence is true?


\item (4 points) For $\alpha \in \R$, let $S_\alpha = (-\alpha,\alpha)$. Prove or disprove the following statements.
\begin{enumerate}
\item $\forall \alpha \in (0,1), \exists \beta \in (0,1), S_\alpha\subset S_\beta$ (note that $\subset$ and $\subseteq$ are not the same).
\item $\exists \alpha \in (0,1), \forall \beta \in (0,1), S_\alpha \subset S_\beta.$
\end{enumerate}

\item (4 points) Let $A, B, C$ be sets. Prove that $(A-B)\cup (A-C)=A-(B\cap C)$. 

\item (4 points) Let $A, B, C$ and $D$ be sets. Prove that
\[
(A\times B)\cap (C\times D)=(A\cap C)\times (B\cap D).
\]

\end{enumerate}

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