66 lines
1.6 KiB
TeX
Executable File
66 lines
1.6 KiB
TeX
Executable File
\section{Boolean Algebra}
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The Kestrel selects its first argument, and the Kite selects its second. This \say{choosing} behavior is similar to something you may have already seen...
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\vspace{1ex}
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Let $T = K\phantom{I} = \lm ab . a$ \par
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Let $F = KI = \lm ab . b$
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\problem{}
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Write a function $\text{NOT}$ so that $(\text{NOT} ~ T) = F$ and $(\text{NOT}~F) = T$. \par
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\hint{What is $(T~\heartsuit~\star)$? How about $(F~\heartsuit~\star)$?}
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\begin{solution}
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$\text{NOT} = \lm a . (a~F~T)$
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\end{solution}
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\vfill
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\problem{}
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Write functions $\text{AND}$, $\text{OR}$, and $\text{XOR}$ that satisfy the following table.
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\begin{center}
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\begin{tabular}{|| c c || c | c | c ||}
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\hline
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$A$ & $B$ & $(\text{AND}~A~B)$ & $(\text{OR}~A~B)$ & $(\text{XOR}~A~B)$ \\
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\hline\hline
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F & F & F & F & F \\
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\hline
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F & T & F & T & T \\
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\hline
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T & F & F & T & T \\
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\hline
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T & T & T & T & F \\
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\hline
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\end{tabular}
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\end{center}
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\begin{solution}
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There's more than one way to do this, of course, but make sure the kids understand how the solutions below work.
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\begin{align*}
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\text{AND} &= \lm ab . (a~b~F) = \lm ab . aba \\
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\text{OR} &= \lm ab . (a~T~b) = \lm ab . aab \\
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\text{XOR} &= \lm ab . (a~ (\text{NOT}~b) ~b)
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\end{align*}
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It may be worth mentioning that OR $= \lm ab.(M~a~b)$ is also a solution.
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\end{solution}
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\vfill
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\problem{}
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To complete our boolean algebra, write the boolean equality check EQ. \par
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What inputs should it take? What outputs should it produce?
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\begin{solution}
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$\text{EQ} = \lm ab . [a~(bTF)~(bFT)] = \lm ab . [a~b~(\text{NOT}~b)]$
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\vspace{1ex}
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$\text{EQ} = \lm ab . [\text{NOT}~(\text{XOR}~a~b)]$
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\end{solution}
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\vfill
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\pagebreak |