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

• spoc21
In summary, the conversation is about a question on proving a statement involving integers x and y. The question asks for tips and suggestions on how to get started, and the conversation also discusses the properties of the set of integers. The attempt at a solution involves using a proof by contradiction, assuming that x and y are integers and y is less than or equal to 0. The summary then briefly explains the steps taken in the attempted solution.
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.

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.

╔(σ_σ)╝ said:
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.

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.

## 1. What is the purpose of solving a problem with sets?

The purpose of solving a problem with sets is to find a solution or set of solutions that satisfy a given set of conditions. This can help to simplify complex problems and make them more manageable.