Finite set exercises.

finite-set-exercises
Joshua Potter 2023-09-20 13:39:59 -06:00
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A \textbf{binary operation} on a set $A$ is a \nameref{ref:function} from
$A \times A$ into $A$.
\section{\defined{Cardinal Arithmetic}}%
\hyperlabel{sec:cardinal-arithmetic}
Let $\kappa$ and $\lambda$ be any cardinal numbers.
\begin{enumerate}[(a)]
\item $\kappa + \lambda = \card{(K \cup L)}$, where $K$ and $L$ are any
disjoint sets of cardinality $\kappa$ and $\lambda$, respectively.
\item $\kappa \cdot \lambda = \card{(K \times L)}$, where $K$ and $L$ are
any sets of cardinality $\kappa$ and $\lambda$, respectively.
\item $\kappa^\lambda = \card{(^L{K})}$, where $K$ and $L$ are any sets of
cardinality $\kappa$ and $\lambda$, respectively.
\end{enumerate}
\lean{Mathlib/SetTheory/Cardinal/Basic}
{Cardinal.add\_def}
\lean{Mathlib/SetTheory/Cardinal/Basic}
{Cardinal.mul\_def}
\lean{Mathlib/SetTheory/Cardinal/Basic}
{Cardinal.power\_def}
\section{\defined{Cardinal Number}}%
\hyperlabel{ref:cardinal-number}
For any set $C$, the \textbf{cardinal number} of set $C$ is denoted as
$\card{C}$.
Furthermore,
\begin{enumerate}[(a)]
\item For any sets $A$ and $B$,
$$\card{A} = \card{B} \quad\text{iff}\quad \equinumerous{A}{B}.$$
\item For a finite set $A$, $\card{A}$ is the \nameref{ref:natural-number}
$n$ for which $\equinumerous{A}{n}$.
\end{enumerate}
\lean{Mathlib/Data/Finset/Card}
{Finset.card}
\lean{Mathlib/SetTheory/Cardinal/Basic}
{Cardinal}
\section{\defined{Cartesian Product}}%
\hyperlabel{ref:cartesian-product}
@ -77,19 +118,6 @@
\lean{Mathlib/Data/Set/Prod}{Set.prod}
\section{\defined{Cardinal Arithmetic}}%
\hyperlabel{sec:cardinal-arithmetic}
Let $\kappa$ and $\lambda$ be any cardinal numbers.
\begin{enumerate}[(a)]
\item $\kappa + \lambda = \card{(K \cup L)}$, where $K$ and $L$ are any
disjoint sets of cardinality $\kappa$ and $\lambda$, respectively.
\item $\kappa \cdot \lambda = \card{(K \times L)}$, where $K$ and $L$ are
any sets of cardinality $\kappa$ and $\lambda$, respectively.
\item $\kappa^\lambda = \card{^L{K}}$, where $K$ and $L$ are any sets of
cardinality $\kappa$ and $\lambda$, respectively.
\end{enumerate}
\section{\defined{Compatible}}%
\hyperlabel{ref:compatible}
@ -1876,7 +1904,7 @@
We proceed by contradiction.
Suppose there existed a set $A$ consisting of every singleton.
Then the \nameref{ref:union-axiom} suggests $\bigcup A$ is a set.
But this set is precisely the class of all sets, which is \textit{not} a
But this "set" is precisely the class of all sets, which is \textit{not} a
set.
Thus our original assumption was incorrect.
That is, there is no set to which every singleton belongs.
@ -9530,7 +9558,7 @@
Refer to \nameref{sub:theorem-6a}.
\end{proof}
\subsection{\sorry{Exercise 6.6}}%
\subsection{\unverified{Exercise 6.6}}%
\hyperlabel{sub:exercise-6.6}
Let $\kappa$ be a nonzero cardinal number.
@ -9538,20 +9566,46 @@
belongs.
\begin{proof}
TODO
Let $\kappa$ be a nonzero cardinal number and define
$$\mathbf{K}_\kappa = \{ X \mid \card{X} = \kappa \}.$$
For the sake of contradiction, suppose $\mathbf{K}_\kappa$ is a set.
Then the \nameref{ref:union-axiom} suggests $\bigcup \mathbf{K}_{\kappa}$ is
a set.
But this "set" is precisely the class of all sets, which is \textit{not} a
set.
Thus our original assumption was incorrect.
That is, there does not exist a set to which every set of cardinality
$\kappa$ belongs.
\end{proof}
\subsection{\sorry{Exercise 6.7}}%
\subsection{\pending{Exercise 6.7}}%
\hyperlabel{sub:exercise-6.7}
Assume that $A$ is finite and $f \colon A \rightarrow A$.
Show that $f$ is one-to-one iff $\ran{f} = A$.
\begin{proof}
TODO
Let $A$ be a \nameref{ref:finite-set} and $f \colon A \rightarrow A$.
\paragraph{($\Rightarrow$)}%
Suppose $f$ is one-to-one.
Then $f$ is a one-to-one correspondence between $A$ and $\ran{f}$.
That is, $\equinumerous{A}{\ran{f}}$.
Because $f$ maps $A$ onto $A$, $\ran{f} \subseteq A$.
Hence $\ran{f} \subset A$ or $\ran{f} = A$.
But, by \nameref{sub:corollary-6c}, $\ran{f}$ cannot be a proper subset of
$A$.
Thus $\ran{f} = A$.
\paragraph{($\Leftarrow$)}%
Suppose $\ran{f} = A$.
TODO
\end{proof}
\subsection{\sorry{Exercise 6.8}}%
\subsection{\pending{Exercise 6.8}}%
\hyperlabel{sub:exercise-6.8}
Prove that the union of two finite sets is finite, without any use of
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TODO
\end{proof}
\subsection{\sorry{Exercise 6.9}}%
\subsection{\pending{Exercise 6.9}}%
\hyperlabel{sub:exercise-6.9}
Prove that the Cartesian product of two finite sets is finite, without any use