# An ordered pair defined as a set

Tags: defined, ordered, pair
 HW Helper P: 3,224 Why can an ordered pair (x, y) be defined as a set {{x}, {x, y}}? Further on, (x, y, z) can de defined as {{x}, {{x}, {{y}, {y, z}}}}... I don't quite understand this.
 PF Patron Sci Advisor Emeritus P: 16,094 What operations can you perform on ordered pairs? Now, interpret those operations in this set-theoretic model. Do those operations satisfy the properties they're supposed to?
 HW Helper P: 3,224 Ok, I think I got it.. There is no order among the elements of a set, hence, since an ordered pair (or n-touple in general) is a set, there has to be a way to imply order in set notation, as well as to keep the fact that (a, b) = (a', b') <=> a=a' & b=b' true. So, from a set {{a}, {a, b}} we can 'read': the set with only one element is {a}, which makes a the first element in the ordered pair (a, b). Assuming a does not equal b, we 'jump' to the next set {a, b}, and select the element b as the second element of (a, b). Analogically, if we have a set { {a}, { {a}, {{b}, {b, c}} } }, we see that the set with one element is {a}, which makes a the first element in (a, b, c). Let's assume a, b and b, c are different. So, we 'jump' to the next set { {a}, {{b}, {b, c}} }. Since, a and b are different, we directly jump to the set {{b}, {b, c}} and select b for the second element of (a, b, c), since {b} is a singleton. And, finally, since b and c are different, we select c for the third element of (a, b, c)... Is this a correct way of thinking?
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