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Mark
2023-05-11 20:05:02 -07:00
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29
Advanced/Definable Sets/parts/2 quantifiers.tex
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@ -8,8 +8,8 @@ $\forall$ and $\exists$ are \textit{quantifiers}. They allow us to make statemen
\vspace{2mm}
Let's look at $\forall$ first. Let $\varphi$ be a formula. \par
Then, the formula $\forall x ~ \varphi$ says \say{$\varphi$ is true for all possible $x$.}
Let's look at $\forall$ first. Let $\varphi(x)$ be a formula. \par
Then, the formula $\forall x ~ \varphi(x)$ says \say{$\varphi$ is true for all possible $x$.}
\vspace{1mm}
@ -18,7 +18,7 @@ In english, this means \say{For any $x$, $x$ is bigger than zero,} or simply \sa
\vspace{3mm}
$\exists$ is very similar: the formula $\exists x ~ \varphi$ states that there is at least one $x$ that makes $\varphi$ true. \par
$\exists$ is very similar: the formula $\exists x ~ \varphi(x)$ states that there is at least one $x$ that makes $\varphi$ true. \par
For example, $\exists ~ (0 < x)$ means \say{there is a positive number in our set}.
\vspace{4mm}
@ -58,6 +58,12 @@ Which are true in $\mathbb{R}^+_0$? \par
\vfill
\pagebreak
\problem{}
Does the order of $\forall$ and $\exists$ in a formula matter? \par
What's the difference between $\exists x ~ \forall y ~ (x < y)$ and $\forall y ~ \exists x ~ (x < y)$? \par
\hint{In $\mathbb{R}^+$, the first is false and the second is true. $\mathbb{R}^+$ does not contain zero.}
\vfill
\problem{}
Define 0 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
@ -71,6 +77,7 @@ Define 1 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
\vfill
\pagebreak
\problem{}
@ -78,13 +85,10 @@ Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
\vfill
\problem{}
Define $3$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
%\problem{}
%Define $2$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
\vfill
\pagebreak
%\vfill
\problem{}
Let $\varphi(x)$ be a formula. \par
@ -94,12 +98,5 @@ Define $(\forall x ~ \varphi(x))$ using logical symbols and $\exists$.
$\Bigl(\forall x ~ \varphi(x)\Bigr)$ is true iff $\lnot \Bigl(\exists x ~ \lnot \varphi(x) \Bigr)$ is true.
\end{solution}
\vfill
\problem{}
Does the order of $\forall$ and $\exists$ in a formula matter? \par
What's the difference between $\exists x ~ \forall y ~ (x < y)$ and $\forall y ~ \exists x ~ (x < y)$? \par
\hint{In $\mathbb{R}^+$, the first is false and the second is true. $\mathbb{R}^+$ does not contain zero.}
\vfill
\pagebreak