Solving a Problem with Sets: x+y <xy, then y>0

[spoc21]

Hi, I'm having a lot of trouble with the following question:

Homework Statement

(a) Let x,y ∈ Z. Prove that if x>0 and x+y <xy, then y>0

Homework Equations

x+y <xy, then y>0

The Attempt at a Solution

I am very confused with this problem, and am not even sure on how to start. Any tips/suggestions to help me get started would be greatly appreciated.

 

[╔(σ_σ)╝]

What properties does Z have ? Is it an ordered field ? A commutative ring , a subset of R etc. Without this information I do not see how we can help you.

 

[Mark44]

What properties does Z have ? Is it an ordered field ? A commutative ring , a subset of R etc. Without this information I do not see how we can help you.

Z is just the set of integers.

 

[Mark44]

I think this might be a way to prove it, using a proof by contradiction.

Assume that x and y are in Z, x + y < xy, and y <= 0.

Since by assumption, y <= 0, then x + y <= x.
Then (x + y)² <= x²
From the above, it follows that y(2x + y) <= 0.

Now, work with that inequality to try to get a contradiction, keeping in mind that x and y can only be integer values, and that x > 0 and y <= 0.