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{\large MATH 108 Fall 2019 \ -  \ {\bf Problem Set 2}}

% Put your name and section here.
{\large due October 11}
\end{center}


\begin{enumerate}
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	\item Let $x$ and $y$ be real numbers.
	\begin{enumerate}
	 \item Prove for all $x$ and $y$ that if $x+y$ is irrational then $x$ is irrational or $y$ is irrational.
	 \item Prove for all $x$ that there exists $y$ such that $x+y$ is rational.
    \end{enumerate}
	
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	\item For all integers $x$, prove that $x$ is divisible by 6 if and only if $x$ is divisible by 2 and by 3.

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	\item
	 \begin{enumerate}
	 \item Prove that there exist integers $m$ and $n$ such that $3m + 4n = 1$.
	 \item Prove that there does not exist integers $m$ and $n$ such that $3m + 6n = 1$.
    \end{enumerate}

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	\item Let $A = \{1,2\}$ and $B = \{1,4,5\}$.
	\begin{enumerate}
	 \item Find $A \cup B$.
	 \item Find $A \cap B$.
	 \item Find $A \setminus B$.
	 \item Find $A \times B$.
	 \item Find $\mathcal{P}(A)$.
    \end{enumerate}

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	\item Let $A,B,C,D$ be sets.  Prove the following propositions.
	\begin{enumerate}
	 \item $(A \cup B) \cap C \subseteq A \cup (B \cap C)$.
	 \item $(A \setminus B) \cap (A \setminus C) = A \setminus (B \cup C)$.
	 \item If $A$ and $B$ are disjoint, then $A \cap C$ and $B \cap C$ are disjoint.
	 \item If $C \subseteq A$ and $D \subseteq B$ then $D \setminus A \subseteq B \setminus C$.
    \end{enumerate}
	
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	\item Let $A$ be the set of positive integers that are not perfect squares.  Let $P$ be the set of prime numbers.  Prove that $P \subseteq A$.
     
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	\item Let $S$ be a set of 4 distinct integers.  Prove that there exists a pair of distinct elements $x,y \in S$ such that $x-y$ is divisible by 3.
     
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\end{enumerate}
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