102 lines
2.9 KiB
TeX
102 lines
2.9 KiB
TeX
\section{Quantifiers}
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Recall the logical symbols we introduced earlier: $(), \land, \lor, \lnot, \rightarrow$ \par
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We will now add two more: $\forall$ (for all) and $\exists$ (exists).
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\definition{}
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$\forall$ and $\exists$ are \textit{quantifiers}. They allow us to make statements about arbitrary symbols.
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\vspace{2mm}
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Let's look at $\forall$ first. Let $\varphi(x)$ be a formula. \par
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Then, the formula $\forall x ~ \varphi(x)$ says \say{$\varphi$ is true for all possible $x$.}
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\vspace{1mm}
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For example, take the formula $\forall x ~ (0 < x)$. \par
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In english, this means \say{For any $x$, $x$ is bigger than zero,} or simply \say{Any $x$ is positive.}
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\vspace{3mm}
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$\exists$ is very similar: the formula $\exists x ~ \varphi(x)$ states that there is at least one $x$ that makes $\varphi$ true. \par
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For example, $\exists ~ (0 < x)$ means \say{there is a positive number in our set}.
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\vspace{4mm}
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\problem{}
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Which of the following are true in $\mathbb{Z}$? \par
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Which are true in $\mathbb{R}^+_0$? \par
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\hint{$\mathbb{R}^+_0$ is the set of positive real numbers and zero.} \par
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\begin{itemize}[itemsep = 1mm]
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\item $\forall x ~ (x \geq 0)$
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\item $\lnot (\exists x ~ (x = 0))$
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\item $\forall x ~ [\exists y ~ (y \times y = x)]$
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\item $\forall xy ~ \exists z ~ (x < z < y)$ \tab \note{This is a compact way to write $\forall x ~ (\forall y ~ (\exists z ~ (x < z < y)))$}
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\item $\lnot \exists x ~ ( \forall y ~ (x < y) )$ %\tab~\tab \note{Solution is below.}
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\end{itemize}
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%\begin{examplesolution}
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% Here is a solution to the last part: $\lnot \exists x ~ ( \forall y ~ (x < y) )$ \par
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%
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% \vspace{4mm}
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%
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% Reading this term-by-term, we get \tab \say{not exists $x$ where (for all $y$ ($x$ smaller than $y$))} \par
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% If we add some grammar, we get \tab \say{There isn't an $x$ where all $y$ are bigger than $x$} \par
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% which we can rephrase as \tab~\tab \say{There isn't a minimum value} \par
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%
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% \vspace{4mm}
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%
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% Which is true in $\mathbb{Z}$ and false in $\mathbb{R}^+_0$
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%\end{examplesolution}
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\vfill
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\pagebreak
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\problem{}
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Does the order of $\forall$ and $\exists$ in a formula matter? \par
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What's the difference between $\exists x ~ \forall y ~ (x < y)$ and $\forall y ~ \exists x ~ (x < y)$? \par
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\hint{In $\mathbb{R}^+$, the first is false and the second is true. $\mathbb{R}^+$ does not contain zero.}
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\vfill
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\problem{}
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Define 0 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
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\vfill
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\problem{}
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Define 1 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
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\vfill
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\pagebreak
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\problem{}
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Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
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\vfill
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%\problem{}
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%Define $2$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
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%\vfill
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\problem{}
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Let $\varphi(x)$ be a formula. \par
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Define $(\forall x ~ \varphi(x))$ using logical symbols and $\exists$.
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\begin{solution}
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$\Bigl(\forall x ~ \varphi(x)\Bigr)$ is true iff $\lnot \Bigl(\exists x ~ \lnot \varphi(x) \Bigr)$ is true.
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\end{solution}
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\vfill
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\pagebreak |