Number of functions between finite sets

Builds on Cardinality of a finite set Rule of product

Statement

Let $A$ and $B$ be finite sets with $|A| = m$ and $|B| = n$ . Let $B^A$ be the set of all functions from $A$ to $B$ . Then $|B^A| = n^m$ .

Remarks

The case $m = 0$ gives $n^0 = 1$ : there is exactly one function from the empty set to $B$ , also when $B$ is empty. A function from $A$ to $B$ is a choice of one value in $B$ for each of the $m$ elements of $A$ . The rule of product then gives $n^m$ . For a subset count, take $B = \{0, 1\}$ . This gives the number of subsets of a finite set.

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

Where this leads

1 page builds on this one.

Number of subsets of a finite set

Tag	`08QQH9F7G0`
URI	`m2://08QQH9F7G0`
Kind	statement, theorem
Writing	W0 stub
MSC	05A05
Status	This copy of the site has no build data. Run `m2 status 08QQH9F7G0`.
Uses	Cardinality of a finite set, Rule of product

Number of functions between finite sets

Statement

Remarks

Reading path

Where this leads

Further reading

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