Edits

Advanced/Definable Sets/parts/2 quantifiers.tex

@@ -58,8 +58,41 @@ Which are true in $\mathbb{R}^+_0$? \par
 \vfill
 \pagebreak
 
+
 \problem{}
-Define $\forall$ using logical symbols and $\exists$
+Define 0 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
+
+
+\vfill
+
+
+\problem{}
+Define 1 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
+
+
+\vfill
+
+
+\problem{}
+Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
+
+\vfill
+
+\problem{}
+Define $3$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
+
+
+
+\vfill
+\pagebreak
+
+\problem{}
+Let $\varphi(x)$ be a formula. \par
+Define $(\forall x ~ \varphi(x))$ using logical symbols and $\exists$.
+
+\begin{solution}
+	$\Bigl(\forall x ~ \varphi(x)\Bigr)$ is true iff $\lnot \Bigl(\exists x ~ \lnot \varphi(x) \Bigr)$ is true.
+\end{solution}
 
 \vfill