How many mappings are there for a finite set of elements?

[mwest]

If S is a finite set having m > 0 elements, how many mappings are there of s into itelf?



I believe there would be however many mappings there are elements> Any suggestions?

 

[jhicks]

I believe there would be however many mappings there are elements

Consider the mapping where every element maps to one particular element (i.e. the constant map). You didn't specify a surjective map that I can tell, so the possibilities are much larger than you currently think.

 

[mwest]

That is the question as to how it is written. Am I anywhere close.

 

[jhicks]

There are more mappings than you have indicated that are surjective, much less not surjective. How many do you think there are that map distinct elements to distinct elements? There are way more than m.

 

[HallsofIvy]

If your set has n members, then you can map the first into any of those n members: you have n choices. The same is true of the second member, etc.

"Fundamental Counting Principle": If you have n choices for A and n choices for B then you have mn choices for A and B.