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\begin{center}
{\large MATH 108 Winter 2019 \ -  \ {\bf Problem Set 2}}

% Put your name and section here.
{\large due January 25}
\end{center}


\begin{enumerate}
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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 For all integers $x$, prove that $x$ is divisible by 6 if and only if $x$ is divisible by both 2 and 3.

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

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	\item Let $A,B,C,D$ be sets.  For each proposition, give a proof or a counterexample.
	\begin{enumerate}
	 \item $(A \cup B) \cap C \subseteq A \cup (B \cap C)$.
	 \item If $A \cap C \subseteq B \cap C$, then $A \subseteq B$.
	 \item $\mathcal{P}(A) \setminus \mathcal{P}(B) \subseteq \mathcal{P}(A \setminus B)$.
	 \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 $D \setminus A \subseteq B \setminus C$.
	 \item If $A \cap B \cap C = \emptyset$, then $A,B,C$ are pair-wise disjoint.
    \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 
	\begin{enumerate}
	 \item Prove that if $U$ and $V$ are finite sets then
      \[ |U| + |V| = |U \cup V| + |U \cap V|. \]
     \item Prove that if $U$ and $V$ are finite sets then $U \cup V$ is finite.
    \end{enumerate}
    
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	\item Prove by example that there exist sets $A$ and $B$ with $A \subsetneq B$ and a function $f:B \to A$ that is injective (1-to-1).

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\end{enumerate}
\end{document}


