First let's define a set of "normal" elements (i.e. not a set of ordered pairs):
$A=\{x\in \mathbb{N}:x>10\}$
Let's call it the "first" definition. I can also define set A and its elements' properties in another way:
$\forall x(x\in A \iff x\in \mathbb{N} \land x>10)$
I can call this the "second" definition, and I know that it is equivalent to the first one.
Now let's try a set of ordered pairs.
$B=\{(x,y)\in \mathbb{N}\times\mathbb{N}:x>y\}$
So far so good. But I am unsure about how to proceed with the second "definition" similiar to the first example. Here is what I have tried:
$\forall x(x \in B \iff x\in \mathbb{N}\times\mathbb{N} \land x_1>x_2)$, where $x_1$ and $x_2$ refer to the first and second element of the ordered pair, though it does feel "informal" to be refering to them in such a way.
$\forall x \forall y((x,y)\in B \iff (x,y)\in \mathbb{N}\times\mathbb{N} \land x>y)$ is another thing I've tried. This one sounds more formal but less intuitive, as I haven't even defined a variable that refers to the element (ordered pair) of set B.
Are my methods correct? Is there something I can improve, or another method I haven't thought of? Am I completely missing the point? Sorry if this is a dumb question, and thank you in advance!
Best Answer
Here's an option:
$\forall x(x \in B \iff \exists x_1\,\exists x_2\,( x_1\in\mathbb N)\land (x_2\in\mathbb N)\land (x=(x_1,x_2)) \land (x_1>x_2))$,