"The ordered pair $(a,b)$ is defined to be the set $\{\{a\},\{a,b\}\}$." ~ Hungerford's Algebra (p.6)
I think this is the first time that i've seen this definition. I've read the wiki page. Is it defined this way, as opposed to a definition relating to functions as in a Cartesian product, because this definition is considered more elementary (or foundational) being that it is related directly to sets?
Also, the definition of an ordered $n$-tuple, according to the wiki page seems vague (perhaps i'm misunderstanding it). For an ordered triple it gives the example:
$(1,2,3) = \{\{(1,2)\},\{(1,2),3\}\}$
but how do we know this is not the ordered pair $((1,2),3)$? Or is the difference between $(1,2,3)$ and $((1,2),3)$ considered trivial?
Thirdly, and perhaps unrelated, what does it mean for a natural number to be defined
$2_{\mathbb{N}} = \{\emptyset,\{\emptyset\} \},$
and is this also done so that we can define $\mathbb{N}$ in terms of sets?