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\begin{center}\textbf{Problem Set 3 \hspace{.7in} Foundations \hspace{.7in} Due 2/23/21} \end{center}

\noindent Four of these problems will be graded (my choice, not yours!), with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor. Submit as a pdf on Moodle.


\vspace{.25in}
The Fibonacci numbers are defined by the two-term recurrence relationship
\[F_1=1, \enspace F_2=1, \enspace F_{n+2}=F_{n+1}+F_n \enspace \mathrm{for} \enspace n \in \mathbb{Z}^+\]
\begin{enumerate} 
\item Find $F_3$ through $F_{10}$.
\item $F_1+F_2+...+F_n=F_{n+2}-1$ for every $n \in \mathbb{Z}^+$.

\vspace{.25in}

\item Find $\mathcal{P}(\{1,2,3\})$.
\item If $A$ has $n$ elements, then $\mathcal{P}(A)$ has {$2^n$} elements. [Hint: Induction!]
\item Take the Set Theory Gateway on WeBWorK.
\item For any sets $A$ and $B$, $(A \bigcup B)' = A' \bigcap B'$. 
\item For any sets $A, B$, and $C$, $A \bigcup (B \bigcap C) = (A \bigcup B) \bigcap (A \bigcup C)$. 
\item For any sets $A, B$, and $C$, $(A - C) \cap B) \subseteq A \cap B$.

\end{enumerate}

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