There is this question about proving the equality of two sets (which comes from each set being a subset of the other):
$S$ = $\{(x,y)$ $\in$ $\mathbb{N}^2: (2-x)(2-y)<2(4-x-y)\}$
$T = \{(1,1), (1,2), (2,1), (1,3), (3,1)\}$
Now I solve $S$'s $x$ condition and get:
$xy < 4$
and I say that this makes it a subset of $T$ because each pair of $T$ has its product of pairs less than $4$. Would this be correct, if not why? Also I proved that $T$ is a subset of $S$ because each of product of its pair satisfies $S$'s condition, which is that the product is less than $4$. Thank you.
You have convinced yourself that $S$ can be rewritten as $$S=\bigl\{(x,y)\in{\mathbb N}^2\bigm| xy<4\bigr\}\ .$$ Since all elements of $T$ are elements of ${\mathbb N}^2$, and obviously fulfill the condition $xy<4$ it is established that $T\subset S$ (and not the converse, as you seem to believe).
In order to complete the proof that $S=T$ it remains to prove that $S\subset T$. This part is missing in your argument: It still could be that there are pairs $(x,y)\in {\mathbb N}^2$ satisfying $xy<4$, which are not listed in $T$. Now when $xy<4$ then at least one of $x$ and $y$ has to be $<2$. (If both $x$ and $y$ are $\geq2$ then their product is $\geq4$.) Therefore all possible pairs in $S$ are of one of the forms $(1,j)$, $(j,1)$, whereby $1\leq j<4$. This implies that indeed $S\subset T$.