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\begin{center}\textbf{Problem Set 3 \hspace{.7in} Foundations \hspace{.7in} Due 2/12/24} \end{center}

\noindent Four of these problems will be graded (our 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 you  must write up your own final submission without reference to any sources other than the textbook and instructor. \\

\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 Do the Chapter 1 Followup assignment on WeBWorK, available via 
\begin{center} https://webwork.coe.edu/webwork2/MTH-215/ \end{center}

\item Consider the formula $1+2+3+...+ n = \frac{n^2+n+1}{2}$.
\begin{enumerate}
	\item Write the formula in sigma notation.
	\item Show that if this formula works for $n = k$, then it also must work for $n = k + 1$.
	\item Explain why mathematical induction does not prove that this formula is true for all $n \in \mathbb N$.
\end{enumerate}


\item Prove that if $A$ has $n$ elements, then $\mathcal{P}(A)$ has {$2^n$} elements. [Hint: Induction!]
\item For any sets $A$ and $B$, $(A \cap B)' = A' \cup B'$. 
\item If $A \subseteq B$ and $C \subseteq D$, then $A \cap C \subseteq B \cap D$.
\item Show that $A - (B \cap A) = (A - B) \cap A$.
\item Show that \[\left(  \bigcup_{i \in I} A_i \right)^\prime = \bigcap_{i \in I} A_i^\prime \] 
\item Show that \[A \cup \bigcap_{i \in I} B_i = \bigcap_{i \in I} \left( A \cup B_i \right)\] 



\end{enumerate}

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