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\title[Math 250 Handout 7 - More problems about sets; Power sets]{Math 250 Handout 7 - More problems about sets; Power sets}\maketitle


\begin{enumerate}
\item Let $A,B,$ and $C$ be sets. Draw a Venn diagram demonstrating each of the following and then prove each of the following.



\begin{enumerate}
\item Prove that $A \cup (B \cap C) = (A\cup B) \cap (A \cup C)$.
\item Prove that $A - (B \cap C) = (A - B) \cup (A - C)$.
\item Prove that $(A - B) \cup (B-A) \subset (A \cup B) - (A \cap B)$.
\item Prove that if $A - B \subset C$, then $ \overline{C} \subset \overline{A} \cup B$.
\item Prove that if $A \subset B$, then $A\cup B - A \cap B = A - B$.
\end{enumerate}
\vspace{1cm}

\item Disprove each of the following statements by giving examples of sets
$A,B,C$ demonstrating that the statement is false. (I.e., give a counterexample.)
\begin{enumerate}
\item For all sets $A,B,C$, $A \cup (B \cap C) = (A\cup B) \cap C$.
\item For all sets $A,B,C$, $A \cap (B \cup C) = (A\cap B) \cup C$.
\item For all sets $A,B,C$, if $A,B$ are subsets of $C$, then $(C-A) - B = C-(A-B)$.
\item For all sets $A,B,C$, if $A,B$ are subsets of $C$ and if $A \subset B$, then $(C-A) \subset (C- B)$.
\end{enumerate}

\newpage
\textbf{Power Sets}\\

\begin{enumerate}
\item Write out 4 elements of
 \begin{enumerate}
 \item $P(\{0,1,2,3,4\})$;
 \item $P(\mathbf{N})$;
 \item $P(\mathbf{R})$.\\
 \end{enumerate}
 \vspace{5cm}

\item Let $A = \{0,1,2\}$. Circle whichever of the following statements
 are true.

 \begin{enumerate}
 \item $\{0\} \subset P(A)$;\vspace{0.3cm}
 \item $\{1,2\} \in P(A)$;\vspace{0.3cm}
 \item $\{\{0,1\},\{1\}\} \subset P(A)$;\vspace{0.3cm}
 \item $\emptyset \in P(A)$;\vspace{0.3cm}
 \item $\emptyset \subset P(A)$;\vspace{0.3cm}
 \item $\{\emptyset\} \in P(A)$. \vspace{0.3cm}
 \item $\{\emptyset\} \subset P(A)$;\vspace{0.3cm}
 \item $\{1,\{1\}\} \subset P(A)$.\\
 \end{enumerate}

\item Let $A$ and $B$ be sets. Prove or disprove:

 \begin{enumerate}
 \item $P(A) \cup P(B) \subset P(A \cup B)$.
 \item $P(A \cup B) \subset P(A) \cup P(B)$.
 \item $P(A) \times P(B) \subset P(A \times B)$.
 \item $A = B$ if and only if $P(A) = P(B)$.
 \end{enumerate}

\end{enumerate}



\end{enumerate}
\end{document}

