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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*}

 \end{document}