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Computer Science 261
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Exercise set \#6
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Due:  Thursday, Oct. 11
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(Incorporating corrections made in class)
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{\bf Set Theory}


Suppose that
\begin{itemize}
\item A = \{a, b, d, e\}
\item B = \{b, c, d, f\}
\item C = \{r, s \}
\end{itemize}
calculate
\begin{itemize}
\item $A \cap B$
\item $A \cup B$
\item $A - B$
\item $C \times A$
\end{itemize}

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{\bf Truth Tables}
Write truth tables for the following:
\begin{itemize}
\item $p \rightarrow q$
\item $\neg p \rightarrow \neg q$
\item $\neg q \rightarrow \neg p$
\item $\neg p \lor q$
\item $\neg (p \land q)$
\item $\neg p \lor \neg q$
\item $\neg (p \lor q)$
\end{itemize}
Digital circuitry uses {\bf nand} (not and) and {\bf nor}
(not or) and inverter gates.  We usually write 0 for false 
(or no current) and 1 for true (or 5v current). With this 
in mind, what will be the output of the circuit given by
\begin{itemize}
\item (p nand q) nor (not p) ?
\item can you draw the circuit?
\item can you construct a circuit that will 
give us $\neg p \lor q$?
\end{itemize}
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{\bf Induction}
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\item{A.} 
\begin{itemize}
\item State the principle of mathematical induction
\item It is a property of positive integers that any subset of 
positive integers has a smallest element in the subset (this is not 
true, for example, of rational numbers or real numbers).  Use this 
fact to give an informal argument for the principle of induction.
{\bf Hint:}  For a given statement about positive integers, 
consider the subset consisting of all positive integers for which
the statement is {\bf false}.
\end{itemize}
\item{B.} Prove the following statements using mathematical induction
\begin{itemize}
\item $\sum_{k=1}^n{k} = \frac{n(n+1)}{2}$
\item $\sum_{k=1}^n{k^2}=\frac{n(2n+1)(n+1)}{6}$
\item $\sum_{k=0}^n{2^k} = 2^{(n+1)}-1$

\item $\sum_{k=0}^n{\frac{1}{2^k}} = 2 - \frac{1}{2^n}$
\item $\sum_{k=1}^n{k2^k}=(n-1)2^{(n+1)}+2$

\item Recall that Fibonacci numbers are defined with 
$F_0 = F_1 = 1, F_2 = 2,\ldots$
and so on.  if $\phi = \frac{(1+\sqrt{5})}{2}$, prove
that $F_n \geq \phi^{{n-1}}$ for $n \geq 1$  You
can use the fact that $\phi^2 = 1+\phi$.  An interesting
exercise would also be to show that $F_n \leq \phi^n$ and 
use this to make a 
guess about $lim_{n \rightarrow \infty}{\frac{F_{n+1}}{F_n}}$.  
(This question was done in class, and so will not be marked)

\end{itemize}
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