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{\Huge
15-151 Homework 5
}

Please submit in class at 8:00am on Thursday 20th July
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\subsection*{Exercises}

\begin{enumerate}[1.]
\item In Question 1, sets $A$, $B$, $C$ and $D$ are defined by:
$$A = \{ n \in \mathbb{N} \mid n \le 64 \}, \quad B = \{ n \in A \mid n \text{ is divisible by } 7 \},$$
$$C = \{ n \in A \mid \exists k \in \mathbb{N},\, n=2^k-1 \}, \quad D = \{ n \in A \mid \exists k \in \mathbb{N},\, n=k^3 \}$$
Write out the following sets in list notation:
\begin{enumerate}[(a)]
\item $B \cap C$;
\points{2}
\item $B \cap D$;
\points{2}
\item $C \cup D$;
\points{2}
\item $(B \setminus C) \cup (C \setminus B)$;
\points{3}
\item $(B \cup C) \setminus (B \cap C)$.
\points{3}
\end{enumerate}

\item Let $X$ and $Y$ be sets.
\begin{enumerate}[(a)]
\item Prove that $(X \setminus Y) \cup (Y \setminus X) = (X \cup Y) \setminus (X \cap Y)$
\points{7}
\item Prove that $X \subseteq Y$ if and only if $Y \setminus (Y \setminus X) = X$.
\points{7}
\end{enumerate}

\item Let $U$ and $V$ be sets. Must it be the true that $\mathcal{P}(U \cup V) = \mathcal{P}(U) \cup \mathcal{P}(V)$? Prove your claim.

\points{9}
\end{enumerate}

\subsection*{Optional but recommended tasks {\normalfont (not for credit)}}

\begin{enumerate}[1.]
\setcounter{enumi}{3}
\item Suppose there is a set of all sets---call this set $\mathcal{S}$. Define a set $B$ by
$$B = \{ X \in \mathcal{S} \mid X \not \in X \}$$
Prove that $B \in B$ if and only if $B \not \in B$. This is nonsense. What went wrong?
\end{enumerate}

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