Exercise 2.1 (The power set of a finite set is a finite $\sigma$-algebra)

Question

Let $\Omega$ be a finite set. Show that the set of all subsets of $\Omega$, $2^\Omega$, is also finite and that it is a $\sigma$-algebra.

Toğrul Topal · Accepted Answer

\begin{enumerate}
	\item $2^\Omega$ is finite.
ewline
	We argue that for an $n$-element set $\Omega$, $n\in\mathbb{N}$, the power set $2^\Omega$ will always have $2^n$ elements. To do so, we induct on $n$.
	\begin{itemize}
		\item 	extit{Induction base.} Let $n=0$. Then $\Omega = \emptyset$ and $2^\Omega = \{\emptyset\}$ with cardinality of $2^0=1$.
		\item 	extit{Induction step.} Let $\Omega$ be a set of cardinality $n+1$. Suppose inductively that we have proven the statement for any set of the cardinality $n$; in particular for $\Omega' := \Omega\setminus\omega_{n+1}$ for some $\omega_{n+1} \in \Omega$. We argue that there are two sets contained in $2^\Omega$: the subsets of $\Omega'$ and the same subsets of $\Omega'$ but also containing $\omega_{n+1}$:
		$$2^\Omega = \left\{E : E \subseteq \Omega' ight\} \bigcup \left\{ E \cup \{\omega_{n+1}\} : E \subseteq \Omega' ight\}.$$
		[$\subseteq$] Let $E \in 2^\Omega$. Then either $\omega_{n+1}
ot\in E$, in which case $E \subseteq \Omega'$ by definition, or $\omega_{n+1}\in E$, in which case $E\setminus\{\omega_{n+1}\} \subseteq \Omega'$ by definition as well. Similarly, we see the [$\supseteq$] part.
ewline
		To summarize, $2^\Omega$ is a union of two disjoint sets, each of cardinality $2^n$. Therefore $\#2^\Omega = 2^n + 2^n = 2\cdot 2^n = 2^{n+1}$.
	\end{itemize}
	\item $2^\Omega$ is a $\sigma$-algebra.
ewline
	\begin{itemize}
		\item We have $\emptyset \in 2^\Omega$ since $\emptyset$ is vacuously a subset of any set $\Omega$.
		\item If $E\subseteq\Omega$, then $\Omega\setminus E \subseteq\Omega$ by definition of set difference.
		\item Finally, if $E_1,E_2,\ldots$ is countable sequence of sets in $\Omega$, then any $x \in \bigcup_{n\in\mathbb{N}} E_n$ must also be contained in some $x\in E_i \subseteq \Omega$ and is thus a subset of $\Omega$.
	\end{itemize}
\end{enumerate}

Exercise 2.1 (The power set of a finite set is a finite $\sigma$-algebra)

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Exercise 2.1 (The power set of a finite set is a finite $\sigma$-algebra)

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