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\flushleft \underline{Homework 2 -- MATH 2510 Spring 2019}\\
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\begin{enumerate}[1.]
\item Determine whether the statement is a tautology, a contradiction, or neither by constructing a truth table.
\begin{enumerate}[a.]
\item $(\neg P \vee Q) \vee (P \wedge Q)$
\item $(\neg P \vee \neg Q) \vee (P \wedge Q)$
\item $P \rightarrow (P \vee Q)$
\item $Q \leftrightarrow (P \wedge Q)$
\end{enumerate}

\item Determine whether the two statements are propositionally equivalent or not by constructing a truth table.
\begin{enumerate}[a.]
\item $P$ and $P \rightarrow P$
\item $P \rightarrow Q$ and $\neg P \rightarrow \neg Q$
\item $P$ and $P \vee \neg P$
\end{enumerate}

\item Determine whether or not the second statement is a propositional consequence of the first statement by constructing a truth table.
\begin{enumerate}[a.]
\item $P \wedge (P \rightarrow Q)$ and $Q$
\item $\neg P \wedge (P \rightarrow Q)$ and $P$
\end{enumerate}

\item Show that $D$ is a propositional consequence of $(D \vee O) \wedge \neg O$.

\item Show that $\neg C$ is a propositional consequence of $(C \rightarrow \neg D) \wedge D)$.

\iffalse
\item Verbally explain why the statement $\exists x [P(x) \wedge \forall y (P(y) \rightarrow y=x)]$ says there is exactly one object for which $P$ is true. (What's intended is an intuitive verbal explanation only).

\item Write a symbolic statement expressing$\ldots$
\begin{enumerate}[a.]
\item ``There are at least three different objects such that $P$."
\item ``There are exactly two different objects such that $P$."
\end{enumerate}

\item Translate into symbols using using appropriate quantifiers and the given predicates.
\begin{enumerate}[a.]
\item Express
\begin{center}
``All differentiable functions are continuous."
\end{center}
with predicates $D(f)$ for ``$f$ is differentiable" and $C(f)$ for ``$f$ is continuous".
\item Express
\begin{center} 
``There is a complex number that is an integer."
\end{center}
with predicates $C(x)$ for ``$x$ is a complex number" and $N(x)$ for ``$x$ is a natural number". 
\item Express 
\begin{center}
``All Lipschitz functions with compact support are absolutely continuous and of bounded variation."
\end{center}
with predicates $L(f)$ for ``$f$ is Lipschitz", $C(f)$ for ``$f$ has compact support", $A(f)$ for ``$f$ is absolutely continuous", and $B(f)$ for ``$f$ is of bounded variation".
\end{enumerate}
\fi

\end{enumerate}

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