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Mockingbird edits

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Mark 2023-03-28 22:16:39 -07:00
parent 9927c8ff43
commit 48ac646ee4
3 changed files with 15 additions and 17 deletions
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4
Advanced/Mock a Mockingbird/parts/00 intro.tex
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@ -18,9 +18,9 @@ As you wander around this forest, you quickly discover two interesting facts:
You also find that this forest has two laws: You also find that this forest has two laws:
\begin{enumerate}[itemsep = 1mm] \begin{enumerate}[itemsep = 1mm]
\item $L_1$, \textit{The Law of Composition}: \\ \item \textit{The Law of Composition}: \\
For any two birds $A$ and $B$, there must be a bird $C$ so that $Cx = A(Bx)$ For any two birds $A$ and $B$, there must be a bird $C$ so that $Cx = A(Bx)$
\item $L_2$, \textit{The Law of the Mockingbird}: \\ \item \textit{The Law of the Mockingbird}: \\
The forest must contain the Mockingbird $M$, which always satisfies $Mx = xx$. \\ The forest must contain the Mockingbird $M$, which always satisfies $Mx = xx$. \\
In other words, the Mockingbird's response to any bird $x$ is the same as $x$'s response to itself. In other words, the Mockingbird's response to any bird $x$ is the same as $x$'s response to itself.
\end{enumerate} \end{enumerate}

15
Advanced/Mock a Mockingbird/parts/01 tmam.tex
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@ -54,13 +54,12 @@ Show that the laws of the forest guarantee that at least one bird is egocentric.
\pagebreak \pagebreak
\problem{} \definition{}
We say a bird $A$ is \textit{agreeable} if for all birds $B$, there is at least one bird $x$ on which $A$ and $B$ agree. \\ We say a bird $A$ is \textit{agreeable} if for all birds $B$, there is at least one bird $x$ on which $A$ and $B$ agree. \\
In other words, $A$ is agreeable if $Ax = Bx$ for some $x$ for all $B$. In other words, $A$ is agreeable if $Ax = Bx$ for some $x$ for all $B$.
\begin{helpbox} \problem{}
\texttt{Def:} $Mx := xx$ Is the Mockingbird agreeable?
\end{helpbox}
\begin{solution} \begin{solution}
We know that $Mx = xx$. \\ We know that $Mx = xx$. \\
@ -115,11 +114,11 @@ Given three arbitrary birds $A$, $B$, and $C$, show that there exists a bird $D$
\lineno{} let A, B, C \lineno{} let A, B, C
\lineno{} \lineno{}
\lineno{} \cmnt{Invoke the Law of Composition:} \lineno{} \cmnt{Invoke the Law of Composition:}
\lineno{} let Q = BC \lineno{} let Qx = B(Cx)
\lineno{} let D = AQ \lineno{} let Dx = A(Qx)
\lineno{} \lineno{}
\lineno{} D = AQ \lineno{} Dx = A(Qx)
\lineno{} = A(BC) \qed{} \lineno{} = A(B(Cx)) \qed{}
\end{alltt} \end{alltt}
\end{solution} \end{solution}

13
Advanced/Mock a Mockingbird/parts/02 kestrel.tex
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@ -16,9 +16,8 @@ Say $A$ is fixated on $B$. Is $A$ fond of $B$?
\begin{solution} \begin{solution}
Yes! See the following proof. Yes! See the following proof.
\begin{alltt} \begin{alltt}
\lineno{} let B \lineno{} let A
\lineno{} let B \lineno{} let B so that Ax = B
\lineno{} let A so that Ax = B
\lineno{} \thus{} AB = B \qed{} \lineno{} \thus{} AB = B \qed{}
\end{alltt} \end{alltt}
\end{solution} \end{solution}
@ -39,8 +38,8 @@ Show that an egocenteric Kestrel is hopelessly egocentric
\begin{solution} \begin{solution}
\begin{alltt} \begin{alltt}
\lineno{} KK = K \lineno{} KK = K
\lineno{} \thus{} (KK)y = K \lineno{} \thus{} (KK)y = K \cmnt{By definition of the Kestrel}
\lineno{} \thus{} Ky = K \qed{} \lineno{} \thus{} Ky = K \qed{} \cmnt{By 01}
\end{alltt} \end{alltt}
\end{solution} \end{solution}
@ -50,7 +49,7 @@ Show that an egocenteric Kestrel is hopelessly egocentric
\problem{} \problem{}
Assume the forest contains a Kestrel. \\ Assume the forest contains a Kestrel. \\
Given $L_1$ and $L_2$, show that at least one bird is hopelessly egocentric. Given the Law of Composition and the Law of the Mockingbird, show that at least one bird is hopelessly egocentric.
\begin{helpbox}[0.75] \begin{helpbox}[0.75]
\texttt{Def:} $K$ is defined by $(Kx)y = x$ \\ \texttt{Def:} $K$ is defined by $(Kx)y = x$ \\
@ -116,7 +115,7 @@ Show that if $K$ is fond of $Kx$, $K$ is fond of $x$.
An egocentric Kestrel must be extremely lonely. Why is this? An egocentric Kestrel must be extremely lonely. Why is this?
\begin{solution} \begin{solution}
If a Kestrel is egocenteric, it must be the only bird in the forest: If a Kestrel is egocenteric, it must be the only bird in the forest!
\begin{alltt} \begin{alltt}
\lineno{} \cmnt{Given} \lineno{} \cmnt{Given}