Add definitions/prompts for Enderton "Axioms and Operations."
parent
aaa7052040
commit
f560006ece
|
|
@ -21,14 +21,14 @@
|
|||
\label{ref:empty-set-axiom}
|
||||
|
||||
There is a set having no members:
|
||||
$$\exists\; B, \forall\; x, x \not\in B.$$
|
||||
$$\exists B, \forall x, x \not\in B.$$
|
||||
|
||||
\section{\defined{Extensionality Axiom}}%
|
||||
\label{ref:extensionality-axiom}
|
||||
|
||||
If two sets have exactly the same members, then they are equal:
|
||||
$$\forall\; A, \forall\; B,
|
||||
\left[\forall\; x, (x \in A \iff x \in B) \Rightarrow A = B\right].$$
|
||||
$$\forall A, \forall B,
|
||||
\left[\forall x, (x \in A \iff x \in B) \Rightarrow A = B\right].$$
|
||||
|
||||
\begin{axiom}
|
||||
|
||||
|
|
@ -36,17 +36,24 @@ If two sets have exactly the same members, then they are equal:
|
|||
|
||||
\end{axiom}
|
||||
|
||||
\section{\partial{Pair Set}}%
|
||||
\label{ref:pair-set}
|
||||
|
||||
For any sets $u$ and $v$, the \textbf{pair set $\{u, v\}$} is the set whose
|
||||
only members are $u$ and $v$.
|
||||
|
||||
\section{\partial{Pairing Axiom}}%
|
||||
\label{ref:pairing-axiom}
|
||||
|
||||
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).$$
|
||||
|
||||
\section{\defined{Powerset}}%
|
||||
\label{ref:powerset}
|
||||
\section{\defined{Power Set}}%
|
||||
\label{ref:power-set}
|
||||
|
||||
The \textbf{powerset} of some set $A$ is the set of all subsets of $A$.
|
||||
For any set $a$, the \textbf{power set $\powerset{a}$} is the set whose members
|
||||
are exactly the subsets of $a$.
|
||||
|
||||
\begin{definition}
|
||||
|
||||
|
|
@ -58,14 +65,28 @@ The \textbf{powerset} of some set $A$ is the set of all subsets of $A$.
|
|||
\label{ref:power-set-axiom}
|
||||
|
||||
For any set $a$, there is a set whose members are exactly the subsets of $a$:
|
||||
$$\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).$$
|
||||
|
||||
\section{\partial{Subset Axioms}}%
|
||||
\label{ref:subset-axioms}
|
||||
|
||||
For each formula $\phi$ not containing $B$, the following is an axiom:
|
||||
$$\forall t_1, \cdots \forall t_k, \forall c,
|
||||
\exists B, \forall x, (x \in B \iff x \in c \land \phi).$$
|
||||
|
||||
\section{\partial{Union Axiom}}%
|
||||
\label{ref:union-axiom}
|
||||
|
||||
For any set $A$, there exists a set $B$ whose elements are exactly the members
|
||||
of the members of $A$:
|
||||
$$\forall A, \exists B, \forall x \left[ x \in B \iff (\exists b \in A) x \in b \right]$$
|
||||
|
||||
\section{\partial{Union Axiom, Preliminary Form}}%
|
||||
\label{ref:union-axiom-preliminary-form}
|
||||
|
||||
For any sets $a$ and $b$, there is a set whose members are those sets belonging
|
||||
either to $a$ or to $b$ (or both):
|
||||
$$\forall\; a, \forall\; b, \exists\; B, \forall\; x,
|
||||
$$\forall a, \forall b, \exists B, \forall x,
|
||||
(x \in B \iff x \in a \text{ or } x \in b).$$
|
||||
|
||||
\endgroup
|
||||
|
|
@ -209,10 +230,10 @@ Show that if $B \subseteq C$, then $\powerset{B} \subseteq \powerset{C}$.
|
|||
{Enderton.Set.Chapter\_1.exercise\_1\_3}
|
||||
|
||||
Let $x \in \powerset{B}$.
|
||||
By definition of the \nameref{ref:powerset}, $x$ is a subset of $B$.
|
||||
By definition of the \nameref{ref:power-set}, $x$ is a subset of $B$.
|
||||
By hypothesis, $B \subseteq C$.
|
||||
Then $x \subseteq C$.
|
||||
Again by definition of the \nameref{ref:powerset}, it follows
|
||||
Again by definition of the \nameref{ref:power-set}, it follows
|
||||
$x \in \powerset{C}$.
|
||||
|
||||
\end{proof}
|
||||
|
|
@ -230,11 +251,11 @@ Show that $\{\{x\}, \{x, y\}\} \in \powerset{\powerset{B}}.$
|
|||
|
||||
Let $x$ and $y$ be members of set $B$.
|
||||
Then $\{x\}$ and $\{x, y\}$ are subsets of $B$.
|
||||
By definition of the \nameref{ref:powerset}, $\{x\}$ and $\{x, y\}$ are
|
||||
By definition of the \nameref{ref:power-set}, $\{x\}$ and $\{x, y\}$ are
|
||||
members of $\powerset{B}$.
|
||||
Then $\{\{x\}, \{x, y\}\}$ is a subset of $\powerset{B}$.
|
||||
By definition of the \nameref{ref:powerset}, $\{\{x\}, \{x, y\}\}$ is a member
|
||||
of $\powerset{\powerset{B}}$.
|
||||
By definition of the \nameref{ref:power-set}, $\{\{x\}, \{x, y\}\}$ is a
|
||||
member of $\powerset{\powerset{B}}$.
|
||||
|
||||
\end{proof}
|
||||
|
||||
|
|
@ -371,4 +392,190 @@ List all the members of $V_4$.
|
|||
|
||||
\end{proof}
|
||||
|
||||
\chapter{Axioms and Operations}%
|
||||
\label{chap:axioms-operations}
|
||||
|
||||
\section{Axioms}%
|
||||
\label{sec:axioms}
|
||||
|
||||
\subsection{\unverified{Theorem 2A}}%
|
||||
\label{sub:theorem-2a}
|
||||
|
||||
\begin{theorem}[2A]
|
||||
|
||||
There is no set to which every set belongs.
|
||||
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\subsection{\unverified{Theorem 2B}}%
|
||||
\label{sub:theorem-2b}
|
||||
|
||||
\begin{theorem}[2B]
|
||||
|
||||
For any nonempty set $A$, there exists a unique set $B$ such that for any
|
||||
$x$, $$x \in B \iff x \text{ belongs to every member of } A.$$
|
||||
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\section{Exercises 3}%
|
||||
\label{sec:exercises-3}
|
||||
|
||||
\subsection{\unverified{Exercise 3.1}}%
|
||||
\label{sub:exercise-3.1}
|
||||
|
||||
Assume that $A$ is the set of integers divisible by $4$.
|
||||
Similarly assume that $B$ and $C$ are the sets of integers divisible by $9$ and
|
||||
$10$, respectively.
|
||||
What is in $A \cap B \cap C$?
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\subsection{\unverified{Exercise 3.2}}%
|
||||
\label{sub:exercise-3.2}
|
||||
|
||||
Give an example of sets $A$ and $B$ for which $\bigcup A = \bigcup B$ but
|
||||
$A \neq B$.
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\subsection{\unverified{Exercise 3.3}}%
|
||||
\label{sub:exercise-3.3}
|
||||
|
||||
Show that every member of a set $A$ is a subset of $\bigcup A$.
|
||||
(This was stated as an example in this section.)
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\subsection{\unverified{Exercise 3.4}}%
|
||||
\label{sub:exercise-3.4}
|
||||
|
||||
Show that if $A \subseteq B$, then $\bigcup A \subseteq \bigcup B$.
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\subsection{\unverified{Exercise 3.5}}%
|
||||
\label{sub:exercise-3.5}
|
||||
|
||||
Assume that every member of $\mathscr{A}$ is a subset of $B$.
|
||||
Show that $\bigcup \mathscr{A} \subseteq B$.
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\subsection{\unverified{Exercise 3.6a}}%
|
||||
\label{sub:exercise-3.6a}
|
||||
|
||||
Show that for any set $A$, $\bigcup \powerset{A} = A$.
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\subsection{\unverified{Exercise 3.6b}}%
|
||||
\label{sub:exercise-3.6b}
|
||||
|
||||
Show that $A \subseteq \powerset{\bigcup A}$.
|
||||
Under what conditions does equality hold?
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\subsection{\unverified{Exercise 3.7a}}%
|
||||
\label{sub:exercise-3.7a}
|
||||
|
||||
Show that for any sets $A$ and $B$,
|
||||
$$\powerset{A} \cap \powerset{B} = \powerset(A \cap B).$$
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\subsection{\unverified{Exercise 3.7b}}%
|
||||
\label{sub:exercise-3.7b}
|
||||
|
||||
Show that $\powerset{A} \cup \powerset{B} \subseteq \powerset(A \cup B)$.
|
||||
Under what conditions does equality hold?
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\subsection{\unverified{Exercise 3.8}}%
|
||||
\label{sub:exercise-3.8}
|
||||
|
||||
Show that there is no set to which every singleton (that is, every set of the
|
||||
form $\{x\}$) belongs.
|
||||
[\textit{Suggestion}: Show that from such a set, we could construct a set to
|
||||
which every set belonged.]
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\subsection{\unverified{Exercise 3.9}}%
|
||||
\label{sub:exercise-3.9}
|
||||
|
||||
Give an example of sets $a$ and $B$ for which $a \in B$ but
|
||||
$\powerset{A} \not\in \powerset{B}$.
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\subsection{\unverified{Exercise 3.10}}%
|
||||
\label{sub:exercise-3.10}
|
||||
|
||||
Show that if $a \in B$, then $\powerset{a} \in \powerset{\powerset{\bigcup B}}$.
|
||||
[\textit{Suggestion}: If you need help, look in the Appendix.]
|
||||
|
||||
\begin{proof}
|
||||
|
||||
TODO
|
||||
|
||||
\end{proof}
|
||||
|
||||
\end{document}
|
||||
|
|
|
|||
|
|
@ -131,7 +131,7 @@
|
|||
\newcommand{\ico}[2]{\left[#1, #2\right)}
|
||||
\newcommand{\ioc}[2]{\left(#1, #2\right]}
|
||||
\newcommand{\ioo}[2]{\left(#1, #2\right)}
|
||||
\newcommand{\powerset}[1]{\mathscr{P}\;#1}
|
||||
\newcommand{\powerset}[1]{\mathscr{P}\,#1}
|
||||
\newcommand{\ubar}[1]{\text{\b{$#1$}}}
|
||||
|
||||
\let\oldemptyset\emptyset
|
||||
|
|
|
|||
Loading…
Reference in New Issue