How to read a set in extension

Question

I need help trying to translate
$\{(i,i) \in N^2\mid i\in N \text{ and } i \leq 10\}$

Im trying to see if this relation is reflexive.

Answer 1

One variant of set-builder notation is the form

$$ \{ f(x) \mid x \in S \} $$

The membership relation for this form

$$ y \in \{ f(x) \mid x \in S \} $$

is defined to be equivalent to the proposition

There exists $x \in S$ such that $y = f(x)$

The given relation is basically of this form; it's maybe interpreted as an example of a slightly more general form

$$ \{ f(x) : x \in S \mid P(x) \} $$

where $P$ is some property that elements of $S$ may have. The membership relation

$$ y \in \{ f(x) : x \in S \mid P(x) \} $$

is simply

There exists $x \in S$ such that $y = f(x)$ and $P(x)$

Your comment is correct; the set

$$ \{(i,i) \in N^2\mid i\in N \text{ and } i \leq 10\} $$

after massaging it into the form I describe above is

$$ \{(i,i) : i\in N \mid i \leq 10\} $$

and is, assuming by $N$ you mean the natural numbers, equal to the set

$$ \{ (0,0), (1,1), \ldots, (10,10) \} $$