Show the set of all functions from A to B is finite

[Samuelb88]

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].

 

[micromass]

Hi Samuelb88!

I think the easiest is to proceed by induction on n. So first show that there are finitely many functions

[tex]\emptyset\rightarrow [m][/tex]

and then show (using the induction hypothesis) that there are finitely many functions

[tex][n+1]\rightarrow [m][/tex]

The fact that this works is because every function [itex]f:[n+1]\rightarrow [m][/tex] restricts to a function [itex]f:[n]\rightarrow [m][/itex].

 

[gb7nash]

Hm, perhaps I'm looking at this from a different angle, but I think the easiest way is the following:

Let A = {x₁,x₂, ... , x_n} and B = {y₁,y₂, ... , x_m}

Step 1: Pick f(x₁)

m ways

Step 2: Pick f(x₂)

m ways

.
.
.

Step n: Pick f(x_n)

m ways.

By the multiplication property...

-------------

If you're not allowed to use combinatorics, I would go with Micromass's method.

 

[Samuelb88]

Alright, proceeding by induction on [itex]n[/itex]. As my base case I cite that for [itex]n=0[/itex], there are no such functions [itex]g: \emptyset \rightarrow [m][/itex] [itex]\Rightarrow[/itex] thus there are finitely many.

*This part I'm a little confused on since either there are [itex]m[/itex] functions (constant functions) such that [itex]g: \emptyset \rightarrow [m][/itex] or there are zero. I believe there zero since there are no elements in the domain of the function [itex]\Rightarrow[/itex] there cannot be any images in the codomain.

As my inductive hypothesis, I suppose that there are finitely many functions [itex]g: [n] \rightarrow [m][/itex]. I want to show there are finitely many functions [itex]g: [n+1] \rightarrow [m][/itex]. By our inductive hypothesis, we know there are finitely many functions from [itex][n] \rightarrow [m][/itex]. Thus let [itex]j \in [n+1][/itex] and define [itex]h: [n+1] \rightarrow [m][/itex] as follows:

[tex]h(j) = \begin{cases}
g(j) & \text{if j not equal to n+1} \\
i & \text{where i in [m], if i = n+1}
\end{cases}[/tex]

Edit: Just realized I didn't quite finish my proof here, although I finished it in my head. :)

At any rate, since there are finitely many [itex]g[/itex] [itex]\Rightarrow[/itex] there are finitely many [itex]h[/itex]. Thus there are finitely many [itex]g: [n+1] \rightarrow [m][/itex]. Done!

 

[Dick]

I don't see why you need induction. As gb7nash pointed out, you can count the functions just by multiplying choices. Why complicate that? And I wouldn't even count that as 'using combinatorics'. It's just multiplying.

 

[TylerH]

You could use the definition of a function (or mapping), which is: A map from A to B is a subset of A x B, where x denotes the cross product. The definition implies that the set of all maps from A to B is the set of all subsets (ie the powerset) of A x B. Then you prove the powerset of A x B is finite.

 

[HallsofIvy]

It should be easy to show that if A has n members and B has m, then there are exactly [itex]m^n[/b] functions from A to B.