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2023-05-14 19:49:13 -07:00
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Advanced/Definable Sets/parts/2 quantifiers.tex
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@ -37,22 +37,22 @@ Which are true in $\mathbb{R}^+_0$? \par
\item $\forall xy ~ \exists z ~ (x < z < y)$ \tab \note{This is a compact way to write $\forall x ~ (\forall y ~ (\exists z ~ (x < z < y)))$}
\item $\lnot \exists x ~ ( \forall y ~ (x < y) )$ \tab~\tab \note{Solution is below.}
\item $\lnot \exists x ~ ( \forall y ~ (x < y) )$ %\tab~\tab \note{Solution is below.}
\end{itemize}
\begin{examplesolution}
Here is a solution to the last part: $\lnot \exists x ~ ( \forall y ~ (x < y) )$ \par
\vspace{4mm}
Reading this term-by-term, we get \tab \say{not exists $x$ where (for all $y$ ($x$ smaller than $y$))} \par
If we add some grammar, we get \tab \say{There isn't an $x$ where all $y$ are bigger than $x$} \par
which we can rephrase as \tab~\tab \say{There isn't a minimum value} \par
\vspace{4mm}
Which is true in $\mathbb{Z}$ and false in $\mathbb{R}^+_0$
\end{examplesolution}
%\begin{examplesolution}
% Here is a solution to the last part: $\lnot \exists x ~ ( \forall y ~ (x < y) )$ \par
%
% \vspace{4mm}
%
% Reading this term-by-term, we get \tab \say{not exists $x$ where (for all $y$ ($x$ smaller than $y$))} \par
% If we add some grammar, we get \tab \say{There isn't an $x$ where all $y$ are bigger than $x$} \par
% which we can rephrase as \tab~\tab \say{There isn't a minimum value} \par
%
% \vspace{4mm}
%
% Which is true in $\mathbb{Z}$ and false in $\mathbb{R}^+_0$
%\end{examplesolution}
\vfill