# Nested Ordered Pair

Tags:
1. Jan 15, 2015

### Prof. 27

1. The problem statement, all variables and given/known data
One way of modeling tuples in set theory is through nested ordered pairs. A notation I'm not familiar with (I'm assuming it means that the following elements are nested into the last one) is used. (a1, a2, a2,... an) = (a1(a2, a3,..., an)). I have never seen the second "(" in the latter part of the equation. My question is what is a nested ordered pair? What are its properties? How can the second element in the latter part of the equation, have lots of elements; while, the whole thing still be an ordered pair? Nested generally means something in something, and an ordered pair is a couple or two tuple, but I can't seem to put the two together.

2. Relevant equations
None that I know of

3. The attempt at a solution
I've spent about 30 minutes searching and I still can't find a definition.

2. Jan 15, 2015

### HallsofIvy

Staff Emeritus
I have never seen that notation before. I have seen the ordered pair defined as the un ordered pair (a, (a, b)).
The point is simply that the un ordered pair (a, (a, b)) tells you that there are two objects, a and b, and that a is being treated differently from b since it is named twice.

3. Jan 15, 2015

### haruspex

I would guess it means as $(a_1, (a_2, (a_3, (...))))$. This is an ordered pair. The second element is an ordered pair. The second element of that is an ordered pair...
Having written that I did a quick search and found http://en.wikipedia.org/wiki/Tuple#Tuples_as_nested_ordered_pairs. This adds the interesting clean-up of making the the innermost ordered pair $(a_n, \phi)$, which is neater than finishing with $(a_{n-1}, a_n)$.