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Finished TMAM hanout

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Mark
2023-03-16 19:36:13 -07:00
parent 9b55f682ce
commit 8d27bf87ce
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103
Advanced/Mock a Mockingbird/parts/01 tmam.tex
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@ -1,11 +1,11 @@
\section{To Mock a Mockingbird}
\problem{}
The bear, a lifelong resident of the forest, tells you that any bird $A$ is fond of at least one other bird. \\
Mark tells you that any bird $A$ is fond of at least one other bird. \\
Complete his proof.
\begin{alltt}
let A \cmnt{Let A be any any bird.}
let Cx := A(Mx) \cmnt{Define C as the composition of A and M}
let Cx = A(Mx) \cmnt{Define C as the composition of A and M}
\cmnt{The rest is up to you.}
CC = ??
@ -20,17 +20,16 @@ Complete his proof.
\begin{solution}
\begin{alltt}
let A \cmnt{Let A be any any bird.}
let Cx := A(Mx) \cmnt{Define C as the composition of A and M}
CC = A(MC)
= A(CC) \qed{}
\lineno{} let A \cmnt{Let A be any any bird.}
\lineno{} let Cx = A(Mx) \cmnt{Define C as the composition of A and M}
\lineno{} CC = A(MC)
\lineno{} = A(CC) \qed{}
\end{alltt}
\end{solution}
\vfill
\problem{}
We say a bird $A$ is \textit{egocentric} if it is fond if itself.
We say a bird $A$ is \textit{egocentric} if it is fond if itself. \\
Show that the laws of the forest guarantee that at least one bird is egocentric.
@ -42,12 +41,12 @@ Show that the laws of the forest guarantee that at least one bird is egocentric.
\begin{solution}
\begin{alltt}
\cmnt{We know M is fond of at least one bird.}
let E so that ME = E
ME = E \cmnt{By definition of fondness}
ME = EE \cmnt{By definition of M}
\thus{} EE = E \qed{}
\lineno{} \cmnt{We know M is fond of at least one bird.}
\lineno{} let E so that ME = E
\lineno{}
\lineno{} ME = E \cmnt{By definition of fondness}
\lineno{} ME = EE \cmnt{By definition of M}
\lineno{} \thus{} EE = E \qed{}
\end{alltt}
\end{solution}
@ -57,7 +56,7 @@ Show that the laws of the forest guarantee that at least one bird is egocentric.
\problem{}
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. \\
This means that $Ax = Bx$.
In other words, $A$ is agreeable if $Ax = Bx$ for some $x$ for all $B$.
\begin{helpbox}
\texttt{Def:} $Mx := xx$
@ -73,15 +72,14 @@ This means that $Ax = Bx$.
\problem{}
Take two birds $A$ and $B$. Let $C$ be their composition. \\
Show that $A$ must be agreeable if $C$ is agreeable. \\
The bear has again given you a hint.
Show that $A$ must be agreeable if $C$ is agreeable.
\begin{alltt}
\cmnt{Given information}
let A, B
let Cx := A(Bx)
let Cx = A(Bx)
let D \cmnt{Arbitrary bird}
let Ex := D(Bx) \cmnt{Define E as the composition of D and B}
let Ex = D(Bx) \cmnt{Define E as the composition of D and B}
Cy = ??
\end{alltt}
@ -93,15 +91,16 @@ The bear has again given you a hint.
\begin{solution}
\begin{alltt}
\cmnt{Given information}
let A, B
let Cx := A(Bx)
let D \cmnt{Arbitrary bird}
let Ex := D(Bx) \cmnt{Define E as the composition of D and B}
Cy = Ey \cmnt{For some y, because C is agreeable}
\thus{} A(By) = Ey
\thus{} A(By) = D(By) \qed{}
\lineno{} \cmnt{Given information}
\lineno{} let A, B
\lineno{} let Cx = A(Bx)
\lineno{}
\lineno{} let D \cmnt{Arbitrary bird}
\lineno{} let Ex = D(Bx) \cmnt{Define E as the composition of D and B}
\lineno{} let y so that Cy = Ey \cmnt{Such a y must exist because C is agreeable}
\lineno{}
\lineno{} A(By) = Ey
\lineno{} = D(By) \qed{}
\end{alltt}
\end{solution}
@ -109,18 +108,18 @@ The bear has again given you a hint.
\pagebreak
\problem{}
Given three arbitrary birds $A$, $B$, and $C$, show that there exists a bird $D$ defined by $Dx = A(B(Cx))$
Given three arbitrary birds $A$, $B$, and $C$, show that there exists a bird $D$ satisfying $Dx = A(B(Cx))$
\begin{solution}
\begin{alltt}
let A, B, C
\cmnt{Invoke the Law of Composition:}
let Q := BC
let D := AQ
D = AQ
= A(BC) \qed{}
\lineno{} let A, B, C
\lineno{}
\lineno{} \cmnt{Invoke the Law of Composition:}
\lineno{} let Q = BC
\lineno{} let D = AQ
\lineno{}
\lineno{} D = AQ
\lineno{} = A(BC) \qed{}
\end{alltt}
\end{solution}
@ -133,7 +132,7 @@ Note that $x$ and $y$ may be the same bird. \\
Show that any two birds in this forest are compatible. \\
\begin{alltt}
let A, B
let Cx := A(Bx)
let Cx = A(Bx)
\end{alltt}
\begin{helpbox}
@ -143,16 +142,16 @@ Show that any two birds in this forest are compatible. \\
\begin{solution}
\begin{alltt}
let A, B
let Cx := A(Bx) \cmnt{Composition}
let y := Cy \cmnt{Let C be fond of y}
Cy = y
\thus{} A(By) = y
let x := By \cmnt{Rename By to x}
Ax = y \qed{}
\lineno{} let A, B
\lineno{}
\lineno{} let Cx = A(Bx) \cmnt{Composition}
\lineno{} let y = Cy \cmnt{Let C be fond of y}
\lineno{}
\lineno{} Cy = y
\lineno{} = A(By)
\lineno{}
\lineno{} let x = By \cmnt{Rename By to x}
\lineno{} Ax = y \qed{}
\end{alltt}
\end{solution}
@ -163,13 +162,13 @@ Show that any bird that is fond of at least one bird is compatible with itself.
\begin{solution}
\begin{alltt}
let A
let x so that Ax = x
Ax = x \qed{}
\lineno{} let A
\lineno{} let x so that Ax = x \cmnt{A is fond of at least one other bird}
\lineno{} Ax = x \qed{}
\end{alltt}
That's it.
\end{solution}
\vfill
\pagebreak