How Many Elements Are in the Semigroup X^X?

[yaganon]

(Moderator's note: thread moved from "Linear & Abstract Algebra")

problem # 12).

Suppose X is a finite set with n elements. Show that the semigroup X^x has n^n elements.

I'm confused. Isn't semigroup a set of functions? So when it says n elements, it actually means n functions? Also what is X^x defined as?

 

[Tinyboss]

problem # 12).

Suppose X is a finite set with n elements. Show that the semigroup X^x has n^n elements.

I'm confused. Isn't semigroup a set of functions? So when it says n elements, it actually means n functions? Also what is X^x defined as?

Well, if this semigroup is isomorphic to the set of functions from X to X, then n^n is clearly the right number. I looked at the semigroup page on Wikipedia and couldn't find that notation, though. Can't find it in your textbook?