A tuple (ordered pair commonly called) is `(x,y)`. The standard set theory definition underpinning this is that `(x,y)` is shorthand for the set of two elements

A tuple (ordered pair commonly called) is `(x,y)`. The standard set theory definition underpinning this is that `(x,y)` is shorthand for the set of two elements `{ {x}, {x, y} }`. Where the "first coordinate" is the set with one element, and the "second coordinate" is the element in the set with two items that is not in the set with one item. It's a standard definition that means we can talk about ordered pairs but still ground it in set theory.