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\begin{document} 
\noindent
{\bf Math 214 \quad Homework 2 \hfill Your name}


\begin{enumerate}

\item (8 points)
Let $P$: $25$ is odd, $Q$: $15$ is prime and $R$: $\frac{1}{3} \in \N$.   State each of the 
following in words, and determine whether they are true or false. 
  \begin{enumerate}
  \item[(a)] $P\vee Q$ 
  \item[(b)] $P\wedge Q$
  \item[(c)] $(\sim{P})\vee Q$
  \item[(d)] $P\wedge (\sim Q \vee R)$
  \end{enumerate}
 
  
\item (8 points) In each of the following, two open sentences $P(x)$ and $Q(x)$ 
over a domain $S$ are given.  For each part, determine $T = \{x \in S : ``P(x)\Rightarrow Q(x)"
\hbox{ is true} \}$ with explanation.
 \begin{enumerate}
 \item $P(x):x-3=5;  Q(x): x> 8;  S=\N$. 
 \item $P(x): x\in [-1,3);  Q(x): x^2\le 4;  S=[-10,10]$.  
 \end{enumerate}


\item (8 points) Let $P, Q$ be statements. Show that $\sim(P\implies Q)$ and $P\wedge (\sim Q)$ 
are logically equivalent using truth table. 

Solution. 
$$\begin{array}{|c|c|c|c|c|}
\hline
P&Q&P\implies Q& \sim(P\implies Q) & P \wedge (\sim Q) \cr
\hline
T & T & & &  \cr
T & F & & &  \cr
F & T & & &  \cr
F & F & & &  \cr
\hline
\end{array}$$


\item (12 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 (8 points) Consider the statement: 

\qquad
``For every integer $n>0$ there is some real number $x>0$ such that $x<1/n$.'' 

(a) Without using words of negation, write a complete sentence that negates the sentence. 

(b) Determine the original statement or its negation is true with explanation. 

\item (16 points) For $\alpha > 0$, let $S_\alpha = (-\alpha,\alpha)$, i.e., the 
open interval with endpoints $-\alpha, \alpha$. 
Prove or disprove the following statements.
(Note that ``$\subset$''  means proper subset.)

\begin{enumerate}
\item $\forall \alpha \in (0,1), \exists \beta \in (0,1), S_\alpha\subset S_\beta$.

[Hint: Find $\beta$ if $\alpha = 0.9, 0.99, 0.999$, etc. and find the 
general rule for specifying $\beta$ for a given $\alpha$.
 
\item $\exists \alpha \in (0,1), \forall \beta \in (0,1), S_\alpha \subset S_\beta.$

[Hint: For $\alpha = 0.1, 0.01, 0.001$, etc.  see whether one has
$S_\alpha \subset S_\beta$ for all $\beta \in (0,1)$. Then ...]

\end{enumerate}
\end{enumerate}
\end{document}

