- #1

Samuelb88

- 162

- 0

## Homework Statement

If A and B are finite, show the set of all functions [itex]f: A \rightarrow B[/itex] is finite.

## Homework Equations

**Lemma.**If A is finite such that [itex]|A|=n[/itex], then there is a bijective correspondence between A and [itex][n][/itex].

*Notation. [itex][n] = \{ 1, ..., n \}[/itex]

## The Attempt at a Solution

Let A, B be finite such that [itex]|A|=n[/itex] and [itex]|B|=m[/itex]. Then by my lemma, I can find bijective maps between both A and [itex][n][/itex], and B and [itex][m][/itex]. Thus in order to show that the set of all functions [itex]f: A \rightarrow B[/itex] is finite, it suffices to show that the set of all functions [itex]g: [n] \rightarrow [m][/itex] is finite.

I'm not quite sure how to proceed from here. Thus far I considered proceeding by cases, that is, when [itex]n=m[/itex], [itex]n<m[/itex], and [itex]m<n[/itex]. If I proceed by cases, I get stuck at the case of [itex]n=m[/itex] because I can't figure out how to show there are finitely many functions that map [itex][n] \rightarrow [m][/itex].