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\title{Math 301: Homework 2}



\begin{document}
\maketitle

\begin{problem}
	Find logical formulae that represent each of the following statements.
	\begin{enumerate}
		\item There is no greatest odd integer.
		\begin{solution}
			
		\end{solution}	 
		
		\item If an integer has a rational square root, then that root is an integer.
		\begin{solution}
			
		\end{solution}	 
		
		\item If the product of two integers is negative, then one of the integers is negative, but not both.
		\begin{solution}
			
		\end{solution}	 
		
		\item There is an integer that is divisible by every integer.
		\begin{solution}
			
		\end{solution}	 
		
	\end{enumerate}
\end{problem}

\begin{problem}
	Find the statements in plain English, involving as few variables as possible, that are represented by each of the following logical formulae.
	\begin{enumerate}
		\item $\forall n\in \mathbb{N}, [(\exists q \in \mathbb{Z}, 4q=n)\Rightarrow  (\exists q\in \mathbb{Z}, 2q=n)]$.
		\begin{solution}
			
		\end{solution}	 
		
		\item $\forall d\in\mathbb{N},[(\exists q\in\mathbb{Z}, n=qd) \Rightarrow (d=1 \vee d=n)]$.
		
		\begin{solution}
			
		\end{solution}	 
		
		\item $\forall a\in\mathbb{R}, [a>0\Rightarrow (\exists b\in\mathbb{R},(b>0\wedge a<b))]$.
		
		\begin{solution}
			
		\end{solution}	 
		
		\item $\exists a\in \mathbb{Z}, [(\exists q\in \mathbb{Z}, 2q=a) \wedge (\exists q\in \mathbb{Z}, 2q+1=a)]$.
		
		\begin{solution}
			
		\end{solution}	 
		
		
	\end{enumerate}
\end{problem}


\begin{problem}
	Prove $$p\vee (q \wedge r) \Rightarrow (p\vee q)\wedge (p\vee r)$$
	by constructing a proof tree whose premise is $p\vee (q \wedge r)$ and whose conclusion is $(p\vee q)\wedge (p\vee r)$.
\end{problem}
\begin{solution}
	
\end{solution}	 

\begin{problem}
	Let $x\in \mathbb{R}$. Prove by contradiction that if $x$ is irrational, then $-x$ and $1/x$, if $x\neq 0$, are irrational.
\end{problem}
\begin{solution}
	
\end{solution}	 

\begin{problem}
	Prove that, for all real numbers $x$ and $y$, if $x$ is irrational, then $x+y$ and $x-y$ are not both rational.
\end{problem}
\begin{solution}
	
\end{solution}	 





\begin{problem} Write the logical formula corresponding to the following statement and then prove it.
	
	\begin{center}
		"There is no smallest positive real number."
	\end{center}
Can you identify the proof techniques and rules of inference that you used in your proof?
	\begin{solution}
		
	\end{solution}	 
	
\end{problem}

\begin{problem}
	Let $X$ be a set and let $p(x)$ be a predicate. Find a logical formula representing the assertion that there are exactly two elements in $X$ for which $p(x)$ is true. Use the structure of this logical formula to prove that there are exactly two real numbers $x$ such that $x^2=1$.
\end{problem}
\begin{solution}
	
\end{solution}	 


\end{document}
