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Definition of an Ordered Pair

  1. Jul 22, 2012 #1
    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?
  2. jcsd
  3. Jul 22, 2012 #2
    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 lets 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.
  4. Jul 22, 2012 #3
    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.
  5. Jul 28, 2012 #4
    No it wouldn't matter, but the first choice is conventional.

    Pretty much from Suppes (1960)

  6. Jul 29, 2012 #5


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    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.
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