Edits

Advanced/Definable Sets/parts/3 sets.tex

@@ -19,8 +19,8 @@ $$
 $$
 
 \definition{Definable Sets}
-Let $S$ be a structure over a language $\mathcal{L}$. \par
-We say a subset $M$ of $\mathcal{L}$ is \textit{definable} if we can write a formula that is true for some $x$ iff $x \in M$.
+Let $S$ be a structure with a universe $U$. \par
+We say a subset $M$ of $U$ is \textit{definable} if we can write a formula that is true for some $x$ iff $x \in M$.
 
 \vspace{4mm}
 
@@ -57,7 +57,7 @@ Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{\text{real}(z)
 \hint{$z$ is nonreal if $x \in \mathbb{C}$ and $x \notin \mathbb{R}$}
 
 \begin{solution}
-	$\Bigl\{ x ~\bigl|~ \lnot (\text{real}(i \times x) = 0) \Bigr\}$
+	$\Bigl\{ x ~\bigl|~ \text{real}(x) = x \Bigr\}$
 \end{solution}
 
 
@@ -65,14 +65,13 @@ Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{\text{real}(z)
 
 
 \problem{}
-Define the set of integers in $\Bigl( \mathbb{R} ~\big|~ \{\times\} \Bigr)$ \par
+Define $\mathbb{R}^+_0$ in $\Bigl( \mathbb{R} ~\big|~ \{\times\} \Bigr)$ \par
 
-\hint{$\text{real}(z)$ gives the real part of a complex number: $\text{real}(3 + 2i) = 3$}
-\hint{$z$ is nonreal if $x \in \mathbb{C}$ and $x \notin \mathbb{R}$}
+\vfill
 
-\begin{solution}
-	$\Bigl\{ x ~\bigl|~ \lnot (\text{real}(i \times x) = 0) \Bigr\}$
-\end{solution}
+\problem{}
+Let $\bigtriangleup$ be a relational symbol. $a \bigtriangleup b$ holds iff $a$ divides $b$. \par
+Define the set of prime numbers in $\Bigl( \mathbb{Z}^+ ~\big|~ \{ \bigtriangleup \} \Bigr)$ \par
 
 \vfill
 
@@ -87,6 +86,38 @@ Define $\mathbb{Z}^+_0$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
 \problem{}
 Define $<$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
 
+\vfill
+\pagebreak
+
+\problem{}
+Consider the structure $S = ( \mathbb{R} ~|~ \{0, \diamond \} )$ \par
+The relation $a \diamond b$ holds if $| a - b | = 1$
+
+\problempart{}
+Define $\{\}$ in $S$.
+
+\problempart{}
+Define ${-1, 1}$ in $S$.
+
+\problempart{}
+Define $\{-2, 2\}$ in $S$.
+
+\vfill
+
+\problem{}
+Let $P$ be the set of all subsets of $\mathbb{Z}^+_0$. This is called a \textit{power set}. \par
+Let $S$ be the stucture $( P ~|~ \{\subseteq\} )$ \par
+
+\problempart{}
+Show that the empty set is definable in $S$.
+
+\problempart{}
+Let $x \Bumpeq y$ be a relation on $P$. $x \Bumpeq y$ holds if $x \cap y \neq \{\}$. \par
+Show that $\Bumpeq$ is definable in $S$.
+
+\problempart{}
+Let $f$ be a function on $P$ defined by $f(x) = \mathbb{Z}^+_0 - x$. This is called the \textit{compliment} of the set $x$. \par
+Show that $f$ is definable in $S$.
 
 \vfill
 \pagebreak