Number of subsets of a finite set

Builds on Cardinality of a finite set Number of functions between finite sets

Statement

Let $X$ be a finite set with $|X| = n$ . Let $\mathcal{P}(X)$ be the power set of $X$ , the set of all subsets of $X$ . Then $|\mathcal{P}(X)| = 2^n$ .

Remarks

The map that sends a subset $S$ to its indicator function $1_S : X \to \{0, 1\}$ is a bijection. Thus the count follows from the number of functions between finite sets. Grouped by size, the count gives the identity in the node sum of binomial coefficients.

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Cardinality of a finite set definition
Rule of product statement
Number of functions between finite sets statement
Number of subsets of a finite set statement

Tag	`0C5RDJE73X`
URI	`m2://0C5RDJE73X`
Kind	statement, theorem
Writing	W0 stub
MSC	05A05
Status	This copy of the site has no build data. Run `m2 status 0C5RDJE73X`.
Uses	Cardinality of a finite set, Number of functions between finite sets

Number of subsets of a finite set

Statement

Remarks

Reading path

Further reading

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