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\flushleft \underline{Homework 5 -- MATH 2510 Spring 2019}\\
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\begin{enumerate}[1.]
\item Prove $\forall x(Ax \rightarrow Cx)$ from the premises $\forall x(Ax \rightarrow Bx)$ and $\forall x (Bx \rightarrow Cx)$.

\item Prove $\exists x(Ax \wedge Cx)$ from the premises $\forall x (Bx \rightarrow Cx)$ and $\exists x (Ax \wedge Bx)$.

\item $\exists y (Cy)$ from the premises $\forall x((Ax \vee Bx) \rightarrow Cx)$ and $\exists y(Ay \wedge Dy)$.

\item Prove $\exists x \neg Ax$ from the premises $(\forall x Ax) \rightarrow (\exists x Bx)$ and $\forall x \neg Bx$. 

\item Prove $\neg \forall x (Ax)$ from the premises $(\exists x \neg Ax) \vee (\exists x \neg Bx)$ and $\forall x Bx$. 

\item Prove $\forall x Bx$ from the premises $(\forall x (Ax \wedge Bx)) \vee (\forall x (Cx \wedge Dx))$ and $\neg \forall x Dx$.

\item Prove $\neg \exists x Ax$ from the premises $(\exists x (Ax \vee Bx)) \rightarrow \forall x Cx$ and $\exists x \neg Cx$. 

\item Translate the statements into symbols and write the proof.
\begin{enumerate}[a.]
\item All beans and peas are legumes. There are no legumes in the garden. Therefore, there are no beans in the garden.
\item There are cucumbers in the garden. If there are any cucumbers in the garden, then there are some pumpkins in the garden. All pumpkins are vegetables. Therefore, there are some vegetables in the garden.
\end{enumerate}

\item Translate the argument into symbols and write the proof:
\begin{center} ``Every philosophical empiricist admires Hume. Some philosophical idealists like no one who admires Hume. Therefore, some philosophical idealists like no philosophical empiricist."\end{center}

\end{enumerate}

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