main.tex

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\begin{document}

\title{Homework 3}

\maketitle

\begin{enumerate}
	\item Give natural deduction proof of the formula
	\begin{displaymath}
		(A \leftrightarrow B) \leftrightarrow (A \to B) \land (\neg A \to \neg B)
	\end{displaymath}

	\item Show that $(A \to B) \to (B \to A)$ cannot be proven. Given example statements 
		for $A$ and $B$ where $A \to B$ is true but $B \to A$ is not. Can you make 
		them mathematical statements?

	\item A formula is in \textit{conjunctive normal form} if it is written as 
	\begin{displaymath}
		X_1 \land X_2 \land \cdots \land X_n 
	\end{displaymath}
	where each 
	\begin{displaymath}
		X_i = Y^1_i \lor \cdots \lor Y^j_i
	\end{displaymath}
	and each $Y^l_i$ is either $\neg A$ or $A$ for some propositional variable $A$. 
	Which of the following are in conjunctive normal form:
	\begin{itemize}
		\item $$ \neg (A \lor B) $$
		\item $$ \neg A \land (B \lor C)$$
		\item $$ \neg A \lor \neg B \land C$$ 
		\item $$ (A \land B) \lor C$$
	\end{itemize}

	\item It is a fact that any formula in propositional logic can be rewritten in 
		conjunctive normal form up to bi-implication. Rewrite the following formula 
		in conjunctive normal form up to bi-implication. 
	\begin{itemize}
		\item $$A \land (B \lor (C \land D))$$ 
		\item $$\neg (A \land (B \lor C))$$
		\item $$(A \to B) \lor C$$ 
	\end{itemize}
		
		
\end{enumerate}

\end{document}