I've got the following problem:
For $M:={0,1,2,3}$ we look at the Monoid $F:=(M^M, \circ)$.
Now I need to state, how many elements F contains. Since every element in a Monoid need to have a neutral element, I get this 6 elements. Is that correct?
Furthermore, I want to state how many elements of F are invertible. Since I can give the inverse of every of these 6 elements, all 6 are invertible. Is this true as well?
Your answers are incorrect. The monoid $M^M$ has $4^4 = 256$ elements. It consists of all functions from the set $\{0, 1, 2, 3\}$ to itself. The inversible elements are the bijections. Therefore there are $4! = 24$ inversible elements in this monoid.