What is the Definition of an Ordered Pair in Set Theory?

[vanmaiden]

I just started studying set theory, and I've seen this definition for an ordered pair

(a,b) = {{a}, {a,b}}

However, I don't understand how this definition makes sense. Could someone explain this definition to me? Maybe use a concrete example too?

 

[SteveL27]

I just started studying set theory, and I've seen this definition for an ordered pair

(a,b) = {{a}, {a,b}}

However, I don't understand how this definition makes sense. Could someone explain this definition to me? Maybe use a concrete example too?

First, you should realize that nobody actually ever uses this definition. We always say that an ordered pair is (x, y) where x and y are elements of some set or sets.

But how can we define this notion purely out of the axioms of set theory? The axioms don't have anything called an ordered pair.

The definition (a,b) = {{a}, {a,b}} has the virtue that given a and b, we can formalize our intuitive notion of an ordered pair as "a is the first item, b is the second item." You can see that the asymmetry of the definition let's us distinguish between (a,b) and (b,a).

Now, having convinced ourselves that we can indeed define ordered pairs using nothing but the axioms of set theory; we can forget all about this definition and just use the intuitive concept of ordered pair. But we always know in the back of our minds that if someone challenged us to prove that there is such a thing as an ordered pair given the axioms of set theory, we can do so.

 

[vanmaiden]

Now, having convinced ourselves that we can indeed define ordered pairs using nothing but the axioms of set theory; we can forget all about this definition and just use the intuitive concept of ordered pair. But we always know in the back of our minds that if someone challenged us to prove that there is such a thing as an ordered pair given the axioms of set theory, we can do so.

I see, but this is a concept that I see on any site that teaches set theory. Would it matter if we wrote {{a}, {a,b}} as {{a,b}, {b}}? Like the fact that we can even represent an ordered pair as {{a}, {a,b}} seems odd.

 

[xxxx0xxxx]

I see, but this is a concept that I see on any site that teaches set theory. Would it matter if we wrote {{a}, {a,b}} as {{a,b}, {b}}? Like the fact that we can even represent an ordered pair as {{a}, {a,b}} seems odd.

No it wouldn't matter, but the first choice is conventional.

Pretty much from Suppes (1960)

Without something like the present definition at hand it is impossible to develop the theory of relations unless the notion of ordered pairs is taken as primitive. Essentially, our only intuition about an ordered pair is that it is an entity representing two objects in a given order..., [The Definition] is adequate with respect to this idea; two ordered pairs are identical only when the first member of one is identical with the first member of the other, and similarly for the two second members.

 

[HallsofIvy]

Essentially either {a, {a,b}} or {b, {a, b}} says that the there are two members and (unlike in just the set {a,b}) distinguishes between them. Whether the "distinguished member" of the set (a in the example, b in the second) is to be the "first" or the "second" in the ordered pair is then a matter of convention.