Minor edits

Advanced/Definable Sets/parts/0 logic.tex

@@ -1,7 +1,7 @@
 \section{Logical Algebra}
 
 \definition{}
-\textit{Logical operators} operate on the values $\{\text{True}, \text{False}\}$, \par
+\textit{Logical operators} operate on the values $\{\texttt{true}, \texttt{false}\}$, \par
 just like algebraic operators operate on numbers. \par
 In this handout, we'll use the following operators:
 \begin{itemize}
@@ -9,7 +9,7 @@ In this handout, we'll use the following operators:
 	\item $\land$: and
 	\item $\lor$: or
 	\item $\rightarrow$: implies
-	\item $()$, parenthesis.
+	\item $()$: parenthesis.
 \end{itemize}
 
 The function of these is defined by \textit{truth tables}:
@@ -19,10 +19,10 @@ The function of these is defined by \textit{truth tables}:
 	\hline
 		$A$ & $B$ & $A \land B$ \\
 	\hline
-	F & F & F \\
-	F & T & F \\
-	T & F & F \\
-	T & T & T
+	\texttt{F} & \texttt{F} & \texttt{F} \\
+	\texttt{F} & \texttt{T} & \texttt{F} \\
+	\texttt{T} & \texttt{F} & \texttt{F} \\
+	\texttt{T}& \texttt{T} & \texttt{T}
 \end{tabular}
 \hfill
 \begin{tabular}{ c | c | c }
@@ -30,10 +30,10 @@ The function of these is defined by \textit{truth tables}:
 	\hline
 		$A$ & $B$ & $A \lor B$ \\
 	\hline
-	F & F & F \\
-	F & T & T \\
-	T & F & T \\
-	T & T & T
+	\texttt{F} & \texttt{F} & \texttt{F} \\
+	\texttt{F} & \texttt{T} & \texttt{T} \\
+	\texttt{T} & \texttt{F} & \texttt{T} \\
+	\texttt{T} & \texttt{T} & \texttt{T}
 \end{tabular}
 \hfill
 \begin{tabular}{ c | c | c }
@@ -41,10 +41,10 @@ The function of these is defined by \textit{truth tables}:
 	\hline
 		$A$ & $B$ & $A \rightarrow B$ \\
 	\hline
-	F & F & T \\
-	F & T & T \\
-	T & F & F \\
-	T & T & T
+	\texttt{F} & \texttt{F} & \texttt{T} \\
+	\texttt{F} & \texttt{T} & \texttt{T} \\
+	\texttt{T} & \texttt{F} & \texttt{F} \\
+	\texttt{T} & \texttt{T} & \texttt{T}
 \end{tabular}
 \hfill
 \begin{tabular}{ c | c }
@@ -52,8 +52,8 @@ The function of these is defined by \textit{truth tables}:
 	\hline
 		$A$ & $\lnot A$ \\
 	\hline
-	T & F \\
-	F & T \\
+	\texttt{T} & \texttt{F} \\
+	\texttt{F} & \texttt{T} \\
 	~ & ~ \\
 	~ & ~ \\
 \end{tabular}
@@ -61,30 +61,40 @@ The function of these is defined by \textit{truth tables}:
 
 \vspace{2mm}
 
-$A \land B$ is only true if both $A$ and $B$ are true. $A \lor B$ is true when $A$ or $B$ (or both) are true. \par
+$A \land B$ is \texttt{true} only if both $A$ and $B$ are \texttt{true}. $A \lor B$ is \texttt{true} if $A$ or $B$ (or both) are \texttt{true}. \par
 $\lnot A$ is the opposite of $A$, which is why it looks like a \say{negative} sign. \par
 
 \vspace{2mm}
 
 $A \rightarrow B$ is a bit harder to understand. Read aloud, this is \say{$A$ implies $B$.} \par
-The only time $\rightarrow$ is false is when $T \rightarrow F$. This may seem counterintuitive, but it will make more sense as we progress through this handout.
+The only time $\rightarrow$ produces \texttt{false} is when $\texttt{true} \rightarrow \texttt{false}$.
+This fact may seem counterintuitive, but will make more sense as we progress through this handout. \par
+\hint{
+	Think about it---if event $\alpha$ implies $\beta$, it is impossible for $\alpha$ to occur without $\beta$. \par
+	This is the only impossibility. All other variants are valid.
+}
 
 \problem{}
 Evaluate the following.
 \begin{itemize}
-	\item $\lnot T$
-	\item $F \lor T$
-	\item $T \land T$
-	\item $(T \land F) \lor T$
-	\item $(T \land F) \lor T$
-	\item $(\lnot (F \lor \lnot T) ) \rightarrow T$
-	\item $(F \rightarrow T) \rightarrow (\lnot F \lor \lnot T)$
+	\item $\lnot \texttt{T}$
+	\item $\texttt{F} \lor \texttt{T}$
+	\item $\texttt{T} \land \texttt{T}$
+	\item $(\texttt{T} \land \texttt{F}) \lor \texttt{T}$
+	\item $(\texttt{T} \land \texttt{F}) \lor \texttt{T}$
+	\item $(\lnot (\texttt{F} \lor \lnot \texttt{T}) ) \rightarrow \texttt{T}$
+	\item $(\texttt{F} \rightarrow \texttt{T}) \rightarrow (\lnot \texttt{F} \lor \lnot \texttt{T})$
 \end{itemize}
 
 \vfill
 \pagebreak
 \begin{instructornote}
-	After the class has done a few definable set problems, you can try to provide some intuition for $\rightarrow$ with the following example.
+	We can also think of $[x \geq 0] \rightarrow b$ as follows:
+	if $x$ isn't the kind of object we care about, we evaluate true and
+	check the next one. If $x$ \textit{is} the kind of object we care about
+	and $b$ is false, we have a counterexample to $[x \geq 0] \rightarrow b$,
+	and thus $\texttt{T} \rightarrow \texttt{F}$ must be false.
+
 
 	\vspace{2mm}
 
@@ -96,17 +106,13 @@ Evaluate the following.
 
 	If $(\text{F} \rightarrow *)$ returned false, statements like the above would be hard to write. \par
 	If $x$ is negative, $\varphi$ doesn't care whether or not it has a root. In this case, $\text{F} \rightarrow *$ must be true to avoid making whole $\forall$ false.
-
-	\vspace{2mm}
-
-	You can think of $[x \geq 0] \rightarrow b$ as a \say{sanity check} in a program: if $x$ isn't the kind of object we care about, return true and check the next one. If $x$ \textit{is} the kind of object we care about and $b$ is false, we have a counterexample to $[x \geq 0] \rightarrow b$, and thus $T \rightarrow F$ must be false.
 \end{instructornote}
 
 
 \problem{}
 Evaluate the following.
 \begin{itemize}
-	\item $A \rightarrow T$ for any $A$
+	\item $A \rightarrow \texttt{T}$ for any $A$
 	\item $(\lnot (A \rightarrow B)) \rightarrow A$ for any $A, B$
 	\item $(A \rightarrow B) \rightarrow (\lnot B \rightarrow \lnot A)$ for any $A, B$
 \end{itemize}
@@ -115,13 +121,14 @@ Evaluate the following.
 
 \problem{}
 Show that $\lnot (A \rightarrow \lnot B)$ is equivalent to $A \land B$. \par
-That is, show that these give the same result for the same $A$ and $B$. \par
+That is, show that these expressions always evaluate to the same value given
+the same $A$ and $B$. \par
 \hint{Use a truth table}
 
 \vfill
 
 \problem{}
-Can you express $A \lor B$ using only $\lnot$, $\rightarrow$, and $()$?
+Write an expression equivalent to $A \lor B$ using only $\lnot$, $\rightarrow$, and $()$?
 
 \begin{solution}
 	$((\lnot A) \rightarrow B)$
@@ -131,6 +138,6 @@ Can you express $A \lor B$ using only $\lnot$, $\rightarrow$, and $()$?
 
 Note that both $\land$ and $\lor$ can be defined using the other logical symbols. \par
 The only logical symbols we \textit{need} are $\lnot$, $\rightarrow$, and $()$. \par
-We include $\land$ and $\lor$ to simplify our logical expressions.
+We include $\land$ and $\lor$ to simplify our expressions.
 
 \pagebreak