r
/
bookshelf
1
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Add lean references to definitions/axioms.

finite-set-exercises
Joshua Potter 2023-05-22 07:06:29 -06:00
parent 8279131d74
commit 88022b723c
2 changed files with 71 additions and 21 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

24
Bookshelf/Apostol.tex
View File

@ -32,7 +32,7 @@ The \textbf{characteristic function} of $S$ is the function $\mathcal{X}_S$ such
\begin{definition} \begin{definition}
\lean{Common/Set/Basic}{Set.characteristic} \lean*{Common/Set/Basic}{Set.characteristic}
\end{definition} \end{definition}
@ -49,7 +49,7 @@ Such a number $B$ is also known as the \textbf{greatest lower bound}.
\begin{definition} \begin{definition}
\lean{Mathlib/Order/Bounds/Basic}{IsGLB} \lean*{Mathlib/Order/Bounds/Basic}{IsGLB}
\end{definition} \end{definition}
@ -148,7 +148,7 @@ A collection of points satisfying \eqref{sec:partition-eq1} is called a
\begin{definition} \begin{definition}
\lean{Common/Set/Partition}{Set.Partition} \lean*{Common/Set/Partition}{Set.Partition}
\end{definition} \end{definition}
@ -180,7 +180,7 @@ That is to say, for each $k = 1, 2, \ldots, n$, there is a real number $s_k$
\begin{definition} \begin{definition}
\lean{Common/Geometry/StepFunction}{Geometry.StepFunction} \lean*{Common/Geometry/StepFunction}{Geometry.StepFunction}
\end{definition} \end{definition}
@ -197,7 +197,7 @@ Such a number $B$ is also known as the \textbf{least upper bound}.
\begin{definition} \begin{definition}
\lean{Mathlib/Order/Bounds/Basic}{IsLUB} \lean*{Mathlib/Order/Bounds/Basic}{IsLUB}
\end{definition} \end{definition}
@ -224,7 +224,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
\begin{axiom} \begin{axiom}
\lean{Mathlib/Data/Real/Basic}{Real.exists\_isLUB} \lean*{Mathlib/Data/Real/Basic}{Real.exists\_isLUB}
\end{axiom} \end{axiom}
@ -676,7 +676,7 @@ For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
\begin{axiom} \begin{axiom}
\leanp{Common/Geometry/Area}{Nonnegative-Property} \leanp*{Common/Geometry/Area}{Nonnegative-Property}
{Nonnegative Property} {Nonnegative Property}
\end{axiom} \end{axiom}
@ -689,7 +689,7 @@ If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
\begin{axiom} \begin{axiom}
\leanp{Common/Geometry/Area}{Additive-Property} \leanp*{Common/Geometry/Area}{Additive-Property}
{Additive Property} {Additive Property}
\end{axiom} \end{axiom}
@ -702,7 +702,7 @@ If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
\begin{axiom} \begin{axiom}
\leanp{Common/Geometry/Area}{Difference-Property} \leanp*{Common/Geometry/Area}{Difference-Property}
{Difference Property} {Difference Property}
\end{axiom} \end{axiom}
@ -715,7 +715,7 @@ If a set $S$ is in $\mathscr{M}$ and if $T$ is congruent to $S$, then $T$ is
\begin{axiom} \begin{axiom}
\leanp{Common/Geometry/Area}{Invariance-Under-Congruence} \leanp*{Common/Geometry/Area}{Invariance-Under-Congruence}
{Invariance Under Congruence} {Invariance Under Congruence}
\end{axiom} \end{axiom}
@ -728,7 +728,7 @@ If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
\begin{axiom} \begin{axiom}
\leanp{Common/Geometry/Area}{Choice-of-Scale} \leanp*{Common/Geometry/Area}{Choice-of-Scale}
{Choice of Scale} {Choice of Scale}
\end{axiom} \end{axiom}
@ -748,7 +748,7 @@ If there is one and only one number $c$ which satisfies the inequalities
\begin{axiom} \begin{axiom}
\leanp{Common/Geometry/Area}{Exhaustion-Property} \leanp*{Common/Geometry/Area}{Exhaustion-Property}
{Exhaustion Property} {Exhaustion Property}
\end{axiom} \end{axiom}

68
Bookshelf/Enderton/Set.tex
View File

@ -17,12 +17,18 @@
\chapter{Reference}% \chapter{Reference}%
\label{chap:reference} \label{chap:reference}
\section{\partial{Empty Set Axiom}}% \section{\defined{Empty Set Axiom}}%
\label{ref:empty-set-axiom} \label{ref:empty-set-axiom}
There is a set having no members: There is a set having no members:
$$\exists B, \forall x, x \not\in B.$$ $$\exists B, \forall x, x \not\in B.$$
\begin{axiom}
\lean*{Mathlib/Init/Set}{Set.emptyCollection}
\end{axiom}
\section{\defined{Extensionality Axiom}}% \section{\defined{Extensionality Axiom}}%
\label{ref:extensionality-axiom} \label{ref:extensionality-axiom}
@ -32,23 +38,43 @@ If two sets have exactly the same members, then they are equal:
\begin{axiom} \begin{axiom}
\lean{Mathlib/Init/Set}{Set.ext} \lean*{Mathlib/Init/Set}{Set.ext}
\end{axiom} \end{axiom}
\section{\partial{Pair Set}}% \section{\defined{Pair Set}}%
\label{ref:pair-set} \label{ref:pair-set}
For any sets $u$ and $v$, the \textbf{pair set $\{u, v\}$} is the set whose For any sets $u$ and $v$, the \textbf{pair set $\{u, v\}$} is the set whose
only members are $u$ and $v$. only members are $u$ and $v$.
\section{\partial{Pairing Axiom}}% \begin{definition}
\ % Add space
\lean*{Mathlib/Init/Set}{Set.insert}
\lean*{Mathlib/Init/Set}{Set.singleton}
\end{definition}
\section{\defined{Pairing Axiom}}%
\label{ref:pairing-axiom} \label{ref:pairing-axiom}
For any sets $u$ and $v$, there is a set having as members just $u$ and $v$: For any sets $u$ and $v$, there is a set having as members just $u$ and $v$:
$$\forall u, \forall v, \exists B, \forall x, $$\forall u, \forall v, \exists B, \forall x,
(x \in B \iff x = u \text{ or } x = v).$$ (x \in B \iff x = u \text{ or } x = v).$$
\begin{axiom}
\ % Add space
\lean*{Mathlib/Init/Set}{Set.insert}
\lean*{Mathlib/Init/Set}{Set.singleton}
\end{axiom}
\section{\defined{Power Set}}% \section{\defined{Power Set}}%
\label{ref:power-set} \label{ref:power-set}
@ -57,31 +83,49 @@ For any set $a$, the \textbf{power set $\powerset{a}$} is the set whose members
\begin{definition} \begin{definition}
\lean{Mathlib/Init/Set}{Set.powerset} \lean*{Mathlib/Init/Set}{Set.powerset}
\end{definition} \end{definition}
\section{\partial{Power Set Axiom}}% \section{\defined{Power Set Axiom}}%
\label{ref:power-set-axiom} \label{ref:power-set-axiom}
For any set $a$, there is a set whose members are exactly the subsets of $a$: For any set $a$, there is a set whose members are exactly the subsets of $a$:
$$\forall a, \exists B, \forall x, (x \in B \iff x \subseteq a).$$ $$\forall a, \exists B, \forall x, (x \in B \iff x \subseteq a).$$
\section{\partial{Subset Axioms}}% \begin{axiom}
\lean*{Mathlib/Init/Set}{Set.powerset}
\end{axiom}
\section{\defined{Subset Axioms}}%
\label{ref:subset-axioms} \label{ref:subset-axioms}
For each formula $\phi$ not containing $B$, the following is an axiom: For each formula $\phi$ not containing $B$, the following is an axiom:
$$\forall t_1, \cdots \forall t_k, \forall c, $$\forall t_1, \cdots \forall t_k, \forall c,
\exists B, \forall x, (x \in B \iff x \in c \land \phi).$$ \exists B, \forall x, (x \in B \iff x \in c \land \phi).$$
\section{\partial{Union Axiom}}% \begin{axiom}
\lean*{Mathlib/Init/Set}{Set.Subset}
\end{axiom}
\section{\defined{Union Axiom}}%
\label{ref:union-axiom} \label{ref:union-axiom}
For any set $A$, there exists a set $B$ whose elements are exactly the members For any set $A$, there exists a set $B$ whose elements are exactly the members
of the members of $A$: of the members of $A$:
$$\forall A, \exists B, \forall x \left[ x \in B \iff (\exists b \in A) x \in b \right]$$ $$\forall A, \exists B, \forall x \left[ x \in B \iff (\exists b \in A) x \in b \right]$$
\section{\partial{Union Axiom, Preliminary Form}}% \begin{axiom}
\lean*{Mathlib/Data/Set/Lattice}{Set.sUnion}
\end{axiom}
\section{\defined{Union Axiom, Preliminary Form}}%
\label{ref:union-axiom-preliminary-form} \label{ref:union-axiom-preliminary-form}
For any sets $a$ and $b$, there is a set whose members are those sets belonging For any sets $a$ and $b$, there is a set whose members are those sets belonging
@ -89,6 +133,12 @@ For any sets $a$ and $b$, there is a set whose members are those sets belonging
$$\forall a, \forall b, \exists B, \forall x, $$\forall a, \forall b, \exists B, \forall x,
(x \in B \iff x \in a \text{ or } x \in b).$$ (x \in B \iff x \in a \text{ or } x \in b).$$
\begin{axiom}
\lean*{Mathlib/Init/Set}{Set.union}
\end{axiom}
\endgroup \endgroup
\chapter{Introduction}% \chapter{Introduction}%