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I was studying Set theory (on a self-study basis, reading O’Leary’s book on logic and set theory and Warner’s book on Set Theory) and I came across the notation for an ordered pair. I was just wondering if that notation is just a convention. I’ve read Eric Steinheart’s explanation in his book More Precisely:

Since sets are not ordered, it might seem like we can't use sets to define ordered pairs. But we can. Of course, the ordered pair (A, B) is not identical with the set {A, B}. It is identical with some more complex set. We want to define a set in which A and B are ordered. We want to define a set that indicates that A comes first and B comes second. We can do this by indicating the two things in the ordered pair and by distinguishing the one that comes first. One way to do this is to distinguish the first thing from the second thing by picking it out and listing it separately alongside the two things. For example: in the set { {A}, {A, B}}, the thing A is listed separately from the two things {A, B}. We can thus identify the ordered pair (A, B) with the set { {A}, {A, B}}. Notice that {{A}, {A, B}} is not identical with { {B}, {A, B} }. Hence (A, B) is not identical with (B,A).

But it doesn’t seem enough. So, I checked Enderton’s book on Set Theory and he gave a detailed explanation, but I don’t get it. Here his explanation: Ordered pair Ordered pair 2

I still can’t see why the notation { {x}, {x, y}} represent an ordered pair (x,y) instead of a single element and a pair. I know this is basic stuff, but I would appreciate if you could help me understand it.

Andrés E. Caicedo
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brigittethecat
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  • You can imagine that ${x,y}$ says the elements of the ordered pair $(x,y)$ and that ${x}$ says that we have to regard $x$ first (or at least that's how I think of it) – 1123581321 May 30 '20 at 17:35
  • That was my first impression, too, but I was unsure if that was the right way to interpret it. – brigittethecat May 30 '20 at 17:39

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There is no one fixed way to define an ordered pair in terms of sets. It is also common to define an ordered pair as $(x,y) := \{x, \{x,y\}\}$. One can prove that $\{x_1, \{x_1,y_1\}\} = \{x_2, \{x_2,y_2\}\} \iff x_1 = x_2$ and $y_1 = y_2$.

The most important point is that it is indeed possible to define an ordered pair in terms of sets, so we are allowed to work with ordered pairs.

Clement Yung
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