Edits

Advanced/Definable Sets/parts/2 quantifiers.tex

@@ -5,7 +5,7 @@ We will now add two more: $\forall$ (for all) and $\exists$ (exists).
 
 \definition{}
 $\forall$ and $\exists$ are \textit{quantifiers}. They allow us to make statements about arbitrary symbols. \par
-\hint{Quantifiers are aptly named: they tell us \textit{how many} symbols satisfy a certain sentence.}
+\note{Quantifiers are aptly named: they tell us \textit{how many} symbols satisfy a certain sentence.}
 
 
 \vspace{2mm}
@@ -26,22 +26,28 @@ For example, $\exists ~ (0 < x)$ means \say{there is a positive number in our se
 \vspace{4mm}
 
 \problem{}
-Which of the following are true in $\mathbb{Z}$? \par
-Which are true in $\mathbb{R}^+_0$? \par
-\hint{$\mathbb{R}^+_0$ is the set of positive real numbers and zero.} \par
+Which of the following are true in $\mathbb{Z}$? Which are true in $\mathbb{R}^+_0$? \par
+\note{$\mathbb{R}^+_0$ is the set of positive real numbers and zero.}
 
 \begin{itemize}[itemsep = 1mm]
-
 	\item $\forall x ~ (x \geq 0)$
 	\item $\lnot (\exists x ~ (x = 0))$
-
 	\item $\forall x ~ [\exists y ~ (y \times y = x)]$
-
-	\item $\forall xy ~ \exists z ~ (x < z < y)$ \tab \note{This is a compact way to write $\forall x ~ (\forall y ~ (\exists z ~ (x < z < y)))$}
-
-	\item $\lnot \exists x ~ ( \forall y ~ (x < y) )$ %\tab~\tab \note{Solution is below.}
+	\item $\forall xy ~ \exists z ~ (x < z < y)$ \tab
+	\note{This is a compact way to write $\forall x ~ (\forall y ~ (\exists z ~ (x < z < y)))$}
+	\item $\lnot \exists x ~ ( \forall y ~ (x < y) )$
 \end{itemize}
 
+\begin{solution}
+	\begin{itemize}
+		\item \say{all $x$ are positive} \tab $\mathbb{R}^+_0$
+		\item \say{zero doesn't exist} \tab neither
+		\item \say{square roots exist} \tab $\mathbb{R}^+_0$
+		\item \say{this set is dense} \tab\null\tab $\mathbb{R}^+_0$
+		\item \say{there is no minimum} \tab $\mathbb{Z}$
+	\end{itemize}
+\end{solution}
+
 %\begin{examplesolution}
 %	Here is a solution to the last part: $\lnot \exists x ~ ( \forall y ~ (x < y) )$ \par
 %
@@ -60,10 +66,22 @@ Which are true in $\mathbb{R}^+_0$? \par
 \vfill
 \pagebreak
 
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+
 \problem{}
 Does the order of $\forall$ and $\exists$ in a formula matter? \par
 What's the difference between $\exists x ~ \forall y ~ (x \leq y)$ and $\forall y ~ \exists x ~ (x \leq y)$? \par
-\hint{In $\mathbb{R}^+$, the first is false and the second is true. $\mathbb{R}^+$ does not contain zero.}
+\hint{
+	Consider $\mathbb{R}^+$\hspace{-1.3ex},\hspace{0.8ex} the set of positive reals. Zero is not positive. \par
+	Which of the above formulas is true in $\mathbb{R}^+$\hspace{-1.3ex},\hspace{0.8ex} and which is false?
+}
 
 \begin{solution}
 	If $\exists x$ is inside $\forall y$, $x$ depends on $y$. We may pick a different value of $x$ for every $y$. \par
@@ -76,6 +94,9 @@ What's the difference between $\exists x ~ \forall y ~ (x \leq y)$ and $\forall
 \problem{}
 Define 0 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
 
+\begin{solution}
+	$\varphi(x) \coloneqq \bigl[~ \forall y ~ x \times y = x ~\bigr]$
+\end{solution}
 
 \vfill
 
@@ -83,6 +104,10 @@ Define 0 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
 \problem{}
 Define 1 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
 
+\begin{solution}
+	$\varphi(x) \coloneqq \bigl[~ \forall y ~ x \times y = y ~\bigr]$
+\end{solution}
+
 
 \vfill
 \pagebreak
@@ -91,6 +116,10 @@ Define 1 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
 \problem{}
 Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
 
+\begin{solution}
+	$\varphi(x) \coloneqq \bigl[~ (x<0) \land \lnot \exists y ~ (x < y < 0) ~\bigr]$
+\end{solution}
+
 \vfill
 
 %\problem{}
@@ -100,10 +129,10 @@ Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
 
 \problem{}
 Let $\varphi(x)$ be a formula. \par
-Write a formula equivalent to $[~ \forall x ~ \varphi(x) ~]$ using only logical symbols and $\exists$.
+Write a formula equivalent to $\forall x ~ \varphi(x)$ using only logical symbols and $\exists$.
 
 \begin{solution}
-	$[~ \forall x ~ \varphi(x) ~]$ is true if and only if $[~ \lnot \exists x ~ \lnot \varphi(x) ~]$ is true.
+	$\forall x ~ \varphi(x)$ is true if and only if $\lnot \exists x ~ \lnot \varphi(x)$ is true.
 \end{solution}
 
 \vfill