# Thread: Proving a set is contain within another set

1. ## Proving two sets are equal

I really don't have any clue as to how to start doing this question...I'd love it if you could help by pointing me in the right direction (and not just solving it for me).

Here`s the question:
S={(x,y)eN^2: (2-x)(2+y)>2(y-x)
T={(1,1),(1,2),(1,3),(2,1),(3,1)}

Prove that S=T. I know that have to prove each set is contained in the other, I just don't know how to go about doing that

2. ## Re: Proving two sets are equal

Note that (2-x)(2+y)>2(y-x) is equivalent to xy < 4.

3. ## Re: Proving two sets are equal

Yeah I'd actually gotten that far by myself and I think I've proven T is a subset of S. But now how do I prove S is a subset of T?

4. ## Re: Proving two sets are equal

Since all numbers are natural (it seems that 0 is not included in natural numbers) and xy < 4, neither x nor y can be >= 4. This leaves only 9 possibilities for the pair (x, y), which can be considered one by one.

5. ## Re: Proving two sets are equal

Ohh I see. Thank you very much for your help