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\title{Theory I, Sheet 9 Solution}

\author{Eugen Sawin}

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\section*{Exercise 9.1.3}

\begin{align*}

ADT_{bool} = &(\Sigma_{bool}, \E)\\

\Sigma_{bool} = &\{true^{(0)}, false^{(0)}, not^{(1)}, and^{(2)}, or^{(2)}\}\\

\E = &\{true \approx not(false),\\

& false \approx not(true),\\

& not(not(x)) \approx x\\

& and(x, y) \approx not(or(not(x), not(y))),\\

& or(x, y) \approx not(and(not(x), not(y)))\}\\

\end{align*}

\section*{Exercise 9.1.4}

\begin{align*}

\D_{bool} &= (M, \J)\\

M &= \{0, 1\}\\

\J(true) &= 1\\

\J(false) &= 0\\

\J(not)(x) &= 1 - x\\

\J(and)(x, y) &= min(x, y)\\

\J(or)(x, y) &= max(x, y)\\

\end{align*}

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