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\flushleft \underline{Homework 4 -- MATH 2510 Spring 2019}\\
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\begin{enumerate}[1.]
\item Prove $C$ from the premises $\neg A \rightarrow (C \wedge D)$, $A \rightarrow B$, and $\neg B$. 

\item Prove $\neg S$ from the premises $P \wedge Q$, $P \rightarrow \neg (Q \wedge R)$, and $S \rightarrow R$. 

\item Prove $B$ from the premises $\neg (A \vee B) \rightarrow C$, $\neg A$, and $\neg C$. 

\item Prove $P \rightarrow Q$ from the premises $P \rightarrow R$ and $P \rightarrow (R \rightarrow Q)$. 

\item Prove $H$ from the premises $\neg S \wedge C$, $W \rightarrow S$, $\neg W \rightarrow T$, and $T \rightarrow H$.

\item Prove $A \rightarrow D$ from the premises $C \rightarrow \neg G$, $A \rightarrow B$, $H \rightarrow D$, $G \vee H$, and $B \leftrightarrow C$. 

\item Prove $F$ from the premises $D, A \rightarrow B, E \rightarrow C, \neg A \rightarrow (D \rightarrow E)$, and $(B \vee C) \rightarrow F$ using the method of proof by contradiction.

\item Translate the English sentence into a sentence involving quantifiers. Clearly define and explain what each predicate you use means.
\begin{enumerate}[a.]
\item All dogs are mammals.
\item Some professors are old.
\item No republican is a democrat.
\item Some integers are not even.
\item Not all matrices are symmetric.
\item Some topologies are metric spaces.
\item Any division algebra that is also associative has dimension 1, have dimension 2, or have dimension 4. (\textit{hint: use different predicates for each statement about dimension})
\end{enumerate}
\end{enumerate}

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