User:Jon Awbrey/Figures and Tables 2

Cactus Language Stretching Exercises Display 1

PNG

LaTeX

\(F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \texttt{(} ~x~ \texttt{,} ~y~ \texttt{)}\)

Cactus Language Stretching Exercises Display 2

PNG

LaTeX

\(\begin{array}{lll} [| \downharpoonleft s \downharpoonright |] & = & [| F |] \\[4pt] & = & F^{-1} (1) \\[4pt] & = & \{~ (x, y) \in \mathbb{B}^2 ~:~ s ~\} \\[4pt] & = & \{~ (x, y) \in \mathbb{B}^2 ~:~ F(x, y) = 1 ~\} \\[4pt] & = & \{~ (x, y) \in \mathbb{B}^2 ~:~ F(x, y) ~\} \\[4pt] & = & \{~ (x, y) \in \mathbb{B}^2 ~:~ \texttt{(} ~x~ \texttt{,} ~y~ \texttt{)} = 1 ~\} \\[4pt] & = & \{~ (x, y) \in \mathbb{B}^2 ~:~ \texttt{(} ~x~ \texttt{,} ~y~ \texttt{)} ~\} \\[4pt] & = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x ~\mathrm{exclusive~or}~ y ~\} \\[4pt] & = & \{~ (x, y) \in \mathbb{B}^2 ~:~ \mathrm{just~one~true~of}~ x, y ~\} \\[4pt] & = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x ~\mathrm{not~equal~to}~ y ~\} \\[4pt] & = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x \nLeftrightarrow y ~\} \\[4pt] & = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x \neq y ~\} \\[4pt] & = & \{~ (x, y) \in \mathbb{B}^2 ~:~ x + y ~\}. \end{array}\)

Cactus Language Stretching Exercises Display 3

LaTeX

\(\begin{matrix} p & = & \upharpoonleft P \upharpoonright & : & X \to \mathbb{B} \\[4pt] q & = & \upharpoonleft Q \upharpoonright & : & X \to \mathbb{B} \\[4pt] (p, q) & = & (\upharpoonleft P \upharpoonright, \upharpoonleft Q \upharpoonright) & : & (X \to \mathbb{B})^2 \end{matrix}\)

Cactus Language Stretching Exercises Display 4

LaTeX

\(\begin{array}{ccccl} F^\$ & = & \underline{(} \ldots, \ldots \underline{)}^\$ & : & (X \to \mathbb{B})^2 \to (X \to \mathbb{B}) \\[4pt] F^\$ (p, q) & = & \underline{(}~p~,~q~\underline{)}^\$ & : & X \to \mathbb{B} \end{array}\)

Cactus Language Stretching Exercises Display 5

LaTeX

\(\begin{matrix} F^\$ (p, q)(x) & = & \underline{(}~p~,~q~\underline{)}^\$ (x) & \in & \mathbb{B} \\[4pt] \Updownarrow & & \Updownarrow \\[4pt] F(p(x), q(x)) & = & \underline{(}~p(x)~,~q(x)~\underline{)} & \in & \mathbb{B} \end{matrix}\)

Cactus Language Stretching Exercises Display 6

LaTeX

\(\begin{array}{lll} [| F^\$ (p, q) |] & = & [| \underline{(}~p~,~q~\underline{)}^\$ |] \\[4pt] & = & (F^\$ (p, q))^{-1} (1) \\[4pt] & = & \{~ x \in X ~:~ F^\$ (p, q)(x) ~\} \\[4pt] & = & \{~ x \in X ~:~ \underline{(}~p~,~q~\underline{)}^\$ (x) ~\} \\[4pt] & = & \{~ x \in X ~:~ \underline{(}~p(x)~,~q(x)~\underline{)} ~\} \\[4pt] & = & \{~ x \in X ~:~ p(x) + q(x) ~\} \\[4pt] & = & \{~ x \in X ~:~ p(x) \neq q(x) ~\} \\[4pt] & = & \{~ x \in X ~:~ \upharpoonleft P \upharpoonright (x) ~\neq~ \upharpoonleft Q \upharpoonright (x) ~\} \\[4pt] & = & \{~ x \in X ~:~ x \in P ~\nLeftrightarrow~ x \in Q ~\} \\[4pt] & = & \{~ x \in X ~:~ x \in P\!-\!Q ~\mathrm{or}~ x \in Q\!-\!P ~\} \\[4pt] & = & \{~ x \in X ~:~ x \in P\!-\!Q ~\cup~ Q\!-\!P ~\} \\[4pt] & = & \{~ x \in X ~:~ x \in P + Q ~\} \\[4pt] & = & P + Q ~\subseteq~ X \\[4pt] & = & [|p|] + [|q|] ~\subseteq~ X \end{array}\)