Post-class edits

Advanced/Definable Sets/parts/3 sets.tex

@@ -32,21 +32,28 @@ Only even numbers satisfy the formula $\varphi(x) = \exists y ~ (y + y = x)$, \p
 So we can define \say{the set of even numbers} as $\{ x ~|~ \exists y ~ (y + y = x) \}$. \par
 Remember---we can only use symbols that are available in our structure!
 
+\problem{}
+Is the empty set definable in any structure?
+
+\vfill
+
+
+\problem{}
+Define $\{0, 1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{<\} \Bigr)$
+
+\begin{instructornote}
+	Here's an interesting fact:
+
+	A finite set of definable elements is always definable. Why? \par
+	An infinite set of definable elements may not be definable.
+\end{instructornote}
+
+\vfill
+
 \problem{}
 Define the set of prime numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\} \Bigr)$
-
 \vfill
 
-
-\problem{}
-Define $\{0\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{<\} \Bigr)$
-
-\vfill
-
-\problem{}
-Define $\{1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{<\} \Bigr)$
-
-\vfill
 \pagebreak
 
 
@@ -84,7 +91,9 @@ Define $\mathbb{Z}^+_0$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
 \vfill
 
 \problem{}
-Define $<$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
+Define $<$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$ \par
+\hint{We can't formally define a relation yet. Don't worry about that for now. \\
+You can repharase this question as \say{given $a,b \in \mathbb{Z}$, can you write a sentence that is true iff $a < b$?}}
 
 \vfill
 \pagebreak
@@ -94,7 +103,7 @@ Consider the structure $S = ( \mathbb{R} ~|~ \{0, \diamond \} )$ \par
 The relation $a \diamond b$ holds if $| a - b | = 1$
 
 \problempart{}
-Define the empty set in $S$.
+Define 0 in $S$.
 
 \problempart{}
 Define $\{-1, 1\}$ in $S$.
@@ -106,10 +115,12 @@ Define $\{-2, 2\}$ in $S$.
 
 \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
+Let $S$ be the stucture $( P ~|~ \{\subseteq\})$ \par
 
 \problempart{}
-Show that the empty set is definable in $S$.
+Show that the empty set is definable in $S$. \par
+\hint{Defining $\{\}$ with $\{x ~|~ x \neq x\}$ is \textbf{not} what we need here. \\
+We need $\varnothing \in P$, the \say{empty set} element in the power set of $\mathbb{Z}^+_0$.}
 
 \problempart{}
 Let $x \Bumpeq y$ be a relation on $P$. $x \Bumpeq y$ holds if $x \cap y \neq \{\}$. \par