Finite set exercises.
parent
2a85d526d7
commit
a368f32558
|
@ -64,6 +64,47 @@
|
||||||
A \textbf{binary operation} on a set $A$ is a \nameref{ref:function} from
|
A \textbf{binary operation} on a set $A$ is a \nameref{ref:function} from
|
||||||
$A \times A$ into $A$.
|
$A \times A$ 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}}%
|
\section{\defined{Cartesian Product}}%
|
||||||
\hyperlabel{ref:cartesian-product}
|
\hyperlabel{ref:cartesian-product}
|
||||||
|
|
||||||
|
@ -77,19 +118,6 @@
|
||||||
|
|
||||||
\lean{Mathlib/Data/Set/Prod}{Set.prod}
|
\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}}%
|
\section{\defined{Compatible}}%
|
||||||
\hyperlabel{ref:compatible}
|
\hyperlabel{ref:compatible}
|
||||||
|
|
||||||
|
@ -1876,7 +1904,7 @@
|
||||||
We proceed by contradiction.
|
We proceed by contradiction.
|
||||||
Suppose there existed a set $A$ consisting of every singleton.
|
Suppose there existed a set $A$ consisting of every singleton.
|
||||||
Then the \nameref{ref:union-axiom} suggests $\bigcup A$ is a set.
|
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.
|
set.
|
||||||
Thus our original assumption was incorrect.
|
Thus our original assumption was incorrect.
|
||||||
That is, there is no set to which every singleton belongs.
|
That is, there is no set to which every singleton belongs.
|
||||||
|
@ -9530,7 +9558,7 @@
|
||||||
Refer to \nameref{sub:theorem-6a}.
|
Refer to \nameref{sub:theorem-6a}.
|
||||||
\end{proof}
|
\end{proof}
|
||||||
|
|
||||||
\subsection{\sorry{Exercise 6.6}}%
|
\subsection{\unverified{Exercise 6.6}}%
|
||||||
\hyperlabel{sub:exercise-6.6}
|
\hyperlabel{sub:exercise-6.6}
|
||||||
|
|
||||||
Let $\kappa$ be a nonzero cardinal number.
|
Let $\kappa$ be a nonzero cardinal number.
|
||||||
|
@ -9538,20 +9566,46 @@
|
||||||
belongs.
|
belongs.
|
||||||
|
|
||||||
\begin{proof}
|
\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}
|
\end{proof}
|
||||||
|
|
||||||
\subsection{\sorry{Exercise 6.7}}%
|
\subsection{\pending{Exercise 6.7}}%
|
||||||
\hyperlabel{sub:exercise-6.7}
|
\hyperlabel{sub:exercise-6.7}
|
||||||
|
|
||||||
Assume that $A$ is finite and $f \colon A \rightarrow A$.
|
Assume that $A$ is finite and $f \colon A \rightarrow A$.
|
||||||
Show that $f$ is one-to-one iff $\ran{f} = A$.
|
Show that $f$ is one-to-one iff $\ran{f} = A$.
|
||||||
|
|
||||||
\begin{proof}
|
\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}
|
\end{proof}
|
||||||
|
|
||||||
\subsection{\sorry{Exercise 6.8}}%
|
\subsection{\pending{Exercise 6.8}}%
|
||||||
\hyperlabel{sub:exercise-6.8}
|
\hyperlabel{sub:exercise-6.8}
|
||||||
|
|
||||||
Prove that the union of two finite sets is finite, without any use of
|
Prove that the union of two finite sets is finite, without any use of
|
||||||
|
@ -9561,7 +9615,7 @@
|
||||||
TODO
|
TODO
|
||||||
\end{proof}
|
\end{proof}
|
||||||
|
|
||||||
\subsection{\sorry{Exercise 6.9}}%
|
\subsection{\pending{Exercise 6.9}}%
|
||||||
\hyperlabel{sub:exercise-6.9}
|
\hyperlabel{sub:exercise-6.9}
|
||||||
|
|
||||||
Prove that the Cartesian product of two finite sets is finite, without any use
|
Prove that the Cartesian product of two finite sets is finite, without any use
|
||||||
|
|
Loading…
Reference in New Issue