Mark/handouts
Mark/handouts
1
0
Fork 0
Code Pull Requests 1 Actions Packages Releases Activity

Edits

Browse Source
This commit is contained in:
Mark 2023-05-11 20:05:02 -07:00
parent 6ea7e9df97
commit b0c9969072
4 changed files with 18 additions and 22 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

3
Advanced/Definable Sets/parts/0 logic.tex
View File

@ -60,7 +60,7 @@ The function of these is defined by \textit{truth tables}:
\vspace{2mm} \vspace{2mm}
$A \land B$ is only true if both $A$ and $B$ are true. $A \lor B$ is only true if $A$ or $B$ (or both) are true. \par $A \land B$ is only true if both $A$ and $B$ are true. $A \lor B$ is true when $A$ or $B$ (or both) are true. \par
$\lnot A$ is the opposite of $A$, which is why it looks like a \say{negative} sign. \par $\lnot A$ is the opposite of $A$, which is why it looks like a \say{negative} sign. \par
\vspace{2mm} \vspace{2mm}
@ -92,6 +92,7 @@ Evaluate the following.
\problem{} \problem{}
Show that $\lnot (A \rightarrow \lnot B)$ is equivalent to $A \land B$. \par Show that $\lnot (A \rightarrow \lnot B)$ is equivalent to $A \land B$. \par
That is, show that these give the same result for the same $A$ and $B$.
\hint{Use a truth table} \hint{Use a truth table}
\vfill \vfill

4
Advanced/Definable Sets/parts/1 structures.tex
View File

@ -9,7 +9,7 @@ A \textit{universe} is a set of meaningless objects. Here are a few examples:
\end{itemize} \end{itemize}
\definition{} \definition{}
A \textit{structure} consists of a universe $U$ and set of symbols. \par A \textit{structure} consists of a universe $U$ and a set of symbols. \par
A structure's symbols give meaning to the objects in its universe. A structure's symbols give meaning to the objects in its universe.
\vspace{2mm} \vspace{2mm}
@ -35,8 +35,6 @@ The equality check $=$ is \textbf{not} a relation symbol. It is included in ever
\example{} \example{}
\def\structgeneric{\ensuremath{}}
The first structure we'll look at is the following: The first structure we'll look at is the following:
$$ $$
\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr) \Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)

29
Advanced/Definable Sets/parts/2 quantifiers.tex
View File

@ -8,8 +8,8 @@ $\forall$ and $\exists$ are \textit{quantifiers}. They allow us to make statemen
\vspace{2mm} \vspace{2mm}
Let's look at $\forall$ first. Let $\varphi$ be a formula. \par Let's look at $\forall$ first. Let $\varphi(x)$ be a formula. \par
Then, the formula $\forall x ~ \varphi$ says \say{$\varphi$ is true for all possible $x$.} Then, the formula $\forall x ~ \varphi(x)$ says \say{$\varphi$ is true for all possible $x$.}
\vspace{1mm} \vspace{1mm}
@ -18,7 +18,7 @@ In english, this means \say{For any $x$, $x$ is bigger than zero,} or simply \sa
\vspace{3mm} \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}. For example, $\exists ~ (0 < x)$ means \say{there is a positive number in our set}.
\vspace{4mm} \vspace{4mm}
@ -58,6 +58,12 @@ Which are true in $\mathbb{R}^+_0$? \par
\vfill \vfill
\pagebreak \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{} \problem{}
Define 0 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$ Define 0 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
@ -71,6 +77,7 @@ Define 1 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
\vfill \vfill
\pagebreak
\problem{} \problem{}
@ -78,13 +85,10 @@ Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
\vfill \vfill
\problem{} %\problem{}
Define $3$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$ %Define $2$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
%\vfill
\vfill
\pagebreak
\problem{} \problem{}
Let $\varphi(x)$ be a formula. \par 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. $\Bigl(\forall x ~ \varphi(x)\Bigr)$ is true iff $\lnot \Bigl(\exists x ~ \lnot \varphi(x) \Bigr)$ is true.
\end{solution} \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 \vfill
\pagebreak \pagebreak

4
Advanced/Definable Sets/parts/3 sets.tex
View File

@ -39,12 +39,12 @@ Define the set of prime numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\
\problem{} \problem{}
Define $\{0\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$ Define $\{0\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{<\} \Bigr)$
\vfill \vfill
\problem{} \problem{}
Define $\{1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$ Define $\{1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{<\} \Bigr)$
\vfill \vfill
\pagebreak \pagebreak