I take it [7] = {1,2,3,4,5,6,7} and [9] = {1,2,3,4,5,6,7,8,9} ...(?)

**Yes this is correct**
why are you adding? you should be multiplying...

**Consider [3] [4] with f(2)=3**
We then have {1,3}

{1,2,3,4}. This gives

but then we have the additional f(2)=3. This would be added on wouldn't it?

I am not familiar with the notation you are using.

?

**No this stands for **
Anyway, I would disagree. We have 8 choices for what to map 3 to. Having made that choice, we again have 8 choices to map the second element to, then 7, then 6, then 5, etc. The total number of functions with this criteria, therefore, is

**Then I would need to multiply by the 8 instead of adding it and would agree except for the final factor of 3**
I agree

what is your definition of

here? Because, unless we restrict the domain somehow, the inverse will technically never be defined.