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

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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.
  • #1
spoc21
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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.
 
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  • #2
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.
 
  • #3
╔(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.
 
  • #4
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.
 

FAQ: Solving a Problem with Sets: x+y <xy, then 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.

2. How is x+y

The inequality x+y

3. Why is it important for y to be greater than 0 in this problem?

Having y be greater than 0 ensures that the solution set does not include any negative numbers, as they would not satisfy the given condition. This helps to narrow down the possible solutions and make the problem more manageable.

4. Can this problem be solved without using sets?

Yes, this problem can be solved without using sets. However, using sets can help to simplify the problem and make it easier to find a solution or set of solutions.

5. How can solving a problem with sets be applied in real life?

Solving a problem with sets can be applied in many real-life situations, such as in mathematics, computer science, and economics. It can also be used in everyday life to solve problems involving multiple variables and conditions.

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