Inverse mapping of a singleton set

[fk378]

1. The problem statement, all variables and given/known data
For X= NxN, Y=N, define the mapping phi: X-->Y as phi(x,y)=x+y. Find the inverse image of phi-inverse (5) of the singleton set {5}. If n: X-->Y is the product operation n(x,y)=xy, find n-inverse (4).


3. The attempt at a solution
I'm not even really sure what the question is asking. If the set is a singleton set, does that mean it sets 5 to itself? So the inverse image would be 5...?

As for the product operation, I don't know how to approach it.

[Dick]

The inverse image of {5} is the set of all things that map to 5 under phi. In the first case all (x,y) in NxN such that x+y=5 and in the second case x*y=4. N is the natural numbers, right?

[fk378]

N is the natural numbers.

So for x+y=5, the inverse image is either x=0,y=5, or x=5,y=0?

And the inverse image for xy=4 is either x=1,y=4, or x=4,y=1?

Do I have to show a proof for it though?

[Dick]

How about x=1,y=4. Doesn't that work in the first case? And is 0 a natural number? I don't think you have all the solutions for the second case either. I'd just start by listing the possibilities. It doesn't seem to me like it's necessary to 'prove' it.