Definitions

ordered-pair

Ordered pair

In mathematics, an ordered pair is a collection of two distinguishable objects, one of which is identified as the first coordinate (or the first entry or left projection) and the other as the second coordinate (second entry, right projection). The usual notation for an ordered pair is (a, b), with first coordinate a and second coordinate b. (This notation could be confused with that of an open interval on the real number line; the variant left langle a,bright rangle can be used to remove this ambiguity.) The pair is "ordered" in the sense that (a, b) is different from (b, a), unless a and b are the same.

Generalities

Let (a1, b1) and (a2, b2) be two ordered pairs. Then the characteristic or defining property of ordered pairs is
(a1, b1) = (a2, b2) ↔ (a1 = a2 & b1 = b2).

Ordered pairs can have other ordered pairs as entries. Hence the ordered pair enables the recursive definition of ordered n-tuples (ordered lists of n terms). For example, the ordered triple (a,b,c) can be defined as (a, (b,c) ), as one pair nested in another. This approach is mirrored in computer programming languages, where it is possible to construct a list of elements from nested ordered pairs. For example, the list (1 2 3 4 5) becomes (1, (2, (3, (4, (5, {} ))))). The Lisp programming language uses such lists as its primary data structure.

The notion of ordered pair is crucial for the definition of Cartesian product and relation.

Set theoretic definitions of the ordered pair

The characteristic property of ordered pairs mentioned in the preceding section is all that is necessary to understand the way ordered pairs are used in the mathematical literature. However, for purposes of foundations of mathematics it has been considered desirable to express the definition of every type of mathematical object in terms of sets, and for ordered pairs this has been done in several ways.

Wiener's definition

Norbert Wiener proposed the first set theoretical definition of the ordered pair in 1914:
(x,y) := {{{x},{}}, { {y} }}.
He observed that this definition would allow all the types of Principia Mathematica to be expressed using sets alone. (In Principia Mathematica, relations of all arities were primitive.)

The standard Kuratowski definition

In axiomatic set theory, the ordered pair (a,b) is usually defined as the Kuratowski pair:
(a,b)K := {{a}, {a,b}}.
The statement that x is the first element of an ordered pair p can then be formulated as
forall{Y}{in}{p}:{x}{in}{Y}
and that x is the second element of p as
(exist{Y}{in}{p}:{x}{in}{Y})and(forall{Y_{1},Y_{2}}{in}{p}:Y_{1}ne Y_{2}rarr ({x}{notin}{Y_{1}}or{x}{notin}{Y_{2}})).
Note that this definition is still valid for the ordered pair p = (x,x) = { {x}, {x,x} } = { {x}, {x} } = { {x} }; in this case the statement (forall{Y_{1},Y_{2}}{in}{p}:Y_{1}ne Y_{2}rarr ({x}{notin}{Y_{1}}or{x}{notin}{Y_{2}})) is trivially true, since it is never the case that Y1Y2.

Variant definitions

The above definition of an ordered pair is "adequate", in the sense that it satisfies the characteristic property that an ordered pair must have (namely: if (a,b)=(x,y), then a=x and b=y), but also arbitrary, as there are many other definitions which are no more complicated and would also be adequate. Examples for other possible definitions include

  1. (a,b)reverse:= { {b}, {a,b} }
  2. (a,b)short:= { a, {a,b} }
  3. (a, b)01:= { {0,a}, {1,b} }

The "reverse" pair is almost never used, as it has no obvious advantages (nor disadvantages) over the usual Kuratowski pair. The "short" pair has the disadvantage that the proof of the characteristic pair property (see above) is more complicated than for the Kuratowski pair (the axiom of regularity has to be used); moreover, as the number 2 is often defined as the set { 0, 1 } = { {}, {0} }, this would mean that 2 is the pair (0,0)short.

Proving the characteristic property of ordered pairs

Prove: (a,b)K = (c,d)K if and only if a=c and b=d.

Kuratowski:

If a=b:

(a,b)K = {{a}, {a,b}} = {{a}, {a,a}} = { {a} },
and (c,d)K = {{c},{c,d}} = { {a} }.
Thus {c} = {c,d} = {a}, and c=d=a=b.

If ab, then {{a}, {a,b}} = {{c},{c,d}}.

Suppose {c,d} = {a}. Then c=d=a, and so {{c},{c,d}} = {{a}, {a,a}} = {{a}, {a}} = { {a} }. But then {{a}, {a,b}} would also equal { {a} }, so b=a, which contradicts ab.

Suppose {c} = {a,b}. Then a=b=c, which contradicts ab.

Therefore {c} = {a}, or c=a, and {c,d} = {a,b}.

If it were true that d=a, then {c,d} = {a,a} = {a} ≠ {a,b}, a contradiction. So d=b. Thus a=c and b=d.

Conversely, if a=c and b=d, then {{a},{a,b} = {{c},{c,d}}. Thus (a,b)K = (c,d)K.

Reverse: (a,b)reverse = {{b},{a,b}} = {{b},{b,a}} = (b,a)K.

If (a,b)reverse = (c,d)reverse, (b,a)K = (d,c)K. Therefore b=d and a=c.

Conversely, if a=c and b=d, then {{b},{a,b}} = {{d},{c,d}}. Thus (a,b)reverse = (c,d)reverse.

Quine-Rosser definition

Rosser (1953) made extensive use of a definition of the ordered pair due to Willard van Orman Quine. The Quine-Rosser definition requires a prior definition of the natural numbers. Let N be the set of natural numbers, and define

varphi(x) = (x setminus N) cup {n+1 : n in (x cap N) }.

Applying this function simply increments every natural number in x. In particular, varphi(x) does not contain the number 0, so that for any sets x and y,

varphi(x) not= {0} cup varphi(y).

Define the ordered pair (A,B) as

(A,B) = {varphi(a) : a in A} cup {varphi(b) cup {0} : b in B }

Extracting all the elements of the pair that do not contain 0 and undoing varphi yields A. Likewise, B can be recovered from the elements of the pair that do contain 0.

This definition of the ordered pair has a single advantage. In type theory, and in set theories such as New Foundations that are outgrowths of type theory, this pair is of the same type as its projections (and hence is termed a "type-level" ordered pair). Hence a function, defined as a set of ordered pairs, has a type only 1 higher than the type of its arguments. For an extensive discussion of ordered pairs in the context of Quinian set theories, see Holmes (1998).

Morse definition

Morse-Kelley set theory, set out in Morse (1965), makes free use of proper classes. Morse defined the ordered pair so as to allow its projections to be proper classes as well as sets. (The Kuratowski definition does not allow this.) He first defined ordered pairs whose projections are sets in Kuratowski's manner. He then redefined the pair (x,y) as (x times {0}) cup (y times {1}), where the component Cartesian products are Kuratowski pairs on sets. This second step renders possible pairs whose projections are proper classes. The Rosser definition in the preceding section also admits proper classes as projections.

Category theory

Product is the category theoretic notion most similar to that of ordered pair. While a number of objects may play the role of pairs, they are all equivalent in the sense of being categorically isomorphic.

References

  • Holmes, Randall, 1998. Elementary Set Theory with a Universal Set. Academia-Bruylant. The publisher has graciously consented to permit diffusion of this monograph via the web. Copyright is reserved.
  • Morse, Anthony P., 1965. A Theory of Sets. Academic Press
  • J. Barkley Rosser, 1953. Logic for mathematicians. McGraw-Hill.

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