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ctor space the entries of an ordered pair can be other ordered pairs enabling the recursive definition of ordered n tuples ordered lists of n objects for example the ordered triple a b c can be defined as a b c i e as one pair nested in another in the ordered pair a b the object a is called the first entry and the object b the second entry of the pair alternatively the objects are called the first and second components the first and second coordinates or the left and right projections of the ordered pair cartesian products and binary relations and hence functions are defined in terms of ordered pairs cf picture generalities edit let a 1 b 1 displaystyle a_ 1 b_ 1 and a 2 b 2 displaystyle a_ 2 b_ 2 be ordered pairs then the characteristic or defining property of the ordered pair is a 1 b 1 a 2 b 2 if and only if a 1 a 2 and b 1 b 2 displaystyle a_ 1 b_ 1 a_ 2 b_ 2 text if and only if a_ 1 a_ 2 text and b_ 1 b_ 2 the set of all ordered pairs whose first entry is in some set a and whose second entry is in some set b is called the cartesian product of a and b and written a b a binary relation between sets a and b is a subset of a b the a b notation may be used for other purposes most notably as denoting open intervals on the real number line in such situations the context will usually make it clear which meaning is intended 1 2 for additional clarification the ordered pair may be denoted by the variant notation a b textstyle langle a b rangle but this notation also has other uses the left and right projection of a pair p is usually denoted by π 1 p and π 2 p or by π ℓ p and π r p respectively in contexts where arbitrary n tuples are considered π n i t is a common notation for the i th component of an n tuple t informal and formal definitions edit in some introductory mathematics textbooks an informal or intuitive definition of ordered pair is given such as for any two objects a and b the ordered pair a b is a notation specifying the two objects a and b in that order 3 this is usually followed by a comparison to a set of two elements pointing out that in a set a and b must be different but in an ordered pair they may be equal and that while the order of listing the elements of a set doesn t matter in an ordered pair changing the order of distinct entries changes the ordered pair this definition is unsatisfactory because it is only descriptive and is based on an intuitive understanding of order however as is sometimes pointed out no harm will come from relying on this description and almost everyone thinks of ordered pairs in this manner 4 a more satisfactory approach is to observe that the characteristic property of ordered pairs given above is all that is required to understand the role of ordered pairs in mathematics hence the ordered pair can be taken as a primitive notion whose associated axiom is the characteristic property this was the approach taken by the n bourbaki group in its theory of sets published in 1954 however this approach also has its drawbacks as both the existence of ordered pairs and their characteristic property must be axiomatically assumed 3 another way to rigorously deal with ordered pairs is to define them formally in the context of set theory this can be done in several ways and has the advantage that existence and the characteristic property can be proven from the axioms that define the set theory one of the most cited versions of this definition is due to kuratowski see below and his definition was used in the second edition of bourbaki s theory of sets published in 1970 even those mathematical textbooks that give an informal definition of ordered pairs will often mention the formal definition of kuratowski in an exercise defining the ordered pair using set theory edit if one agrees that set theory is an appealing foundation of mathematics then all mathematical objects must be defined as sets of some sort hence if the ordered pair is not taken as primitive it must be defined as a set 5 several set theoretic definitions of the ordered pair are given below see also diepert 6 wiener s definition edit norbert wiener proposed the first set theoretical definition of the ordered pair in 1914 7 a b a b displaystyle left a b right left left left a right emptyset right left left b right right right he observed that this definition made it possible to define the types of principia mathematica as sets principia mathematica had taken types and hence relations of all arities as primitive wiener used b instead of b to make the definition compatible with type theory where all elements in a class must be of the same type with b nested within an additional set its type is equal to a displaystyle a emptyset s hausdorff s definition edit about the same time as wiener 1914 felix hausdorff proposed his definition a b a 1 b 2 displaystyle a b left a 1 b 2 right where 1 and 2 are two distinct objects different from a and b 8 kuratowski s definition edit in 1921 kazimierz kuratowski offered the now accepted definition 9 10 of the ordered pair a b a b k a a b displaystyle a b _ k a a b when the first and the second coordinates are identical the definition obtains x x k x x x x x x displaystyle x x _ k x x x x x x given some ordered pair p the property x is the first coordinate of p can be formulated as y p x y displaystyle forall y in p x in y the property x is the second coordinate of p can be formulated as y p x y y 1 y 2 p x y 1 x y 2 y 1 y 2 displaystyle exists y in p x in y land forall y_ 1 y_ 2 in p x in y_ 1 land x in y_ 2 rightarrow y_ 1 y_ 2 in the case that the left and right coordinates are identical the right conjunct y 1 y 2 p x y 1 x y 2 y 1 y 2 displaystyle forall y_ 1 y_ 2 in p x in y_ 1 land x in y_ 2 rightarrow y_ 1 y_ 2 is trivially true since y 1 y 2 displaystyle y_ 1 y_ 2 is the case if p x y x x y displaystyle p x y x x y then p x x y x x y x displaystyle bigcap p bigcap bigg x x y bigg x cap x y x p x x y x x y x y displaystyle bigcup p bigcup bigg x x y bigg x cup x y x y this is how we can extract the first coordinate of a pair using the iterated operation notation for arbitrary intersection and arbitrary union π 1 p p x x displaystyle pi _ 1 p bigcup bigcap p bigcup x x this is how the second coordinate can be extracted π 2 p a p p p a p a x y x y x a x y y displaystyle pi _ 2 p bigcup left left a in bigcup p right bigcup p neq bigcap p rightarrow a notin bigcap p right bigcup left left a in x y right x y neq x rightarrow a notin x right bigcup y y if x y displaystyle x neq y then the set y displaystyle y could be obtained more simply y a x y a x displaystyle y left a in x y right a notin x but the previous formula also takes into account the case when x y displaystyle x y note that π 1 displaystyle pi _ 1 and π 2 displaystyle pi _ 2 are generalized functions in the sense that their domains and codomains are proper classes variants edit the above kuratowski definition of the ordered pair is adequate in that it satisfies the characteristic property that an ordered pair must satisfy namely that a b x y a x b y displaystyle a b x y leftrightarrow a x land b y in particular it adequately expresses order in that a b b a displaystyle a b b a is false unless b a displaystyle b a there are other definitions of similar or lesser complexity that are equally adequate a b reverse b a b displaystyle a b _ text reverse b a b a b short a a b displaystyle a b _ text short a a b a b 01 0 a 1 b displaystyle a b _ text 01 0 a 1 b 11 the reverse definition is merely a trivial variant of the kuratowski definition and as such is of no independent interest the definition short is so called because it requires two rather than three pairs of braces proving that short satisfies the characteristic property requires the zermelo fraenkel set theory axiom of regularity 12 moreover if one uses von neumann s set theoretic construction of the natural numbers then 2 is defined as the set 0 1 0 0 which is indistinguishable from the pair 0 0 short yet another disadvantage of the short pair is the fact that even if a and b are of the same type the elements of the short pair are not however if a b then the short version keeps having cardinality 2 which is something one might expect of any pair including any ordered pair proving that definitions satisfy the characteristic property edit prove a b c d if and only if a c and b d kuratowski if if a c and b d then a a b c c d thus a b k c d k only if two cases a b and a b if a b a b k a a b a a a a c c d c d k a b k a thus c c d a which implies a c and a d by hypothesis a b hence b d if a b then a b k c d k implies 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 that b a which contradicts a b suppose c a b then a b c which also contradicts a b therefore c a so that c a and c d a b if d a were true then c d a a a a b a contradiction thus d b is the case so that a c and b d reverse a b reverse b a b b b a b a k if if a b reverse c d reverse b a k d c k therefore b d and a c only if if a c and b d then b a b d c d thus a b reverse c d reverse short 13 if if a c and b d then a a b c c d thus a b short c d short only if suppose a a b c c d then a is in the left hand side and thus in the right hand side because equal sets have equal elements one of a c or a c d must be the case if a c d then by similar reasoning as above a b is in the right hand side so a b c or a b c d if a b c then c is in c d a and a is in c and this combination contradicts the axiom of regularity as a c has no minimal element under the relation element of if a b c d then a is an element of a from a c d a b again contradicting regularity hence a c must hold again we see that a b c or a b c d the option a b c and a c implies that c is an element of c contradicting regularity so we have a c and a b c d and so b a b a c d c d so b d quine rosser definition edit rosser 1953 14 employed a definition of the ordered pair due to quine which requires a prior definition of the natural numbers let n displaystyle mathbb n be the set of natural numbers and define first σ x x if x n x 1 if x n displaystyle sigma x begin cases x text if x notin mathbb n x 1 text if x in mathbb n end cases the function σ displaystyle sigma increments its argument if it is a natural number and leaves it as is otherwise the number 0 does not appear in the range of σ displaystyle sigma as x n displaystyle x setminus mathbb n is the set of the elements of x displaystyle x not in n displaystyle mathbb n go on with φ x σ x σ α α x x n n 1 n x n displaystyle varphi x sigma x sigma alpha mid alpha in x x setminus mathbb n cup n 1 n in x cap mathbb n this is the set image of a set x displaystyle x under σ displaystyle sigma sometimes denoted by σ x displaystyle sigma x as well applying function φ displaystyle varphi to a set x simply increments every natural number in it in particular φ x displaystyle varphi x never contains the number 0 so that for any sets x and y φ x 0 φ y displaystyle varphi x neq 0 cup varphi y further define ψ x σ x 0 φ x 0 displaystyle psi x sigma x cup 0 varphi x cup 0 by this ψ x displaystyle psi x does always contain the number 0 finally define the ordered pair a b as the disjoint union a b φ a ψ b φ a a a φ b 0 b b displaystyle a b varphi a cup psi b varphi a a in a cup varphi b cup 0 b in b which is φ a ψ b displaystyle varphi a cup psi b in alternate notation extracting all the elements of the pair that do not contain 0 and undoing φ displaystyle varphi yields a likewise b can be recovered from the elements of the pair that do contain 0 15 for example the pair a 0 b c 1 d 2 e f 3 displaystyle a 0 b c 1 d 2 e f 3 is encoded as a 1 b c 2 d 3 0 e f 4 0 displaystyle a 1 b c 2 d 3 0 e f 4 0 provided a b c d e f n displaystyle a b c d e f notin mathbb n in type theory and in outgrowths thereof such as the axiomatic set theory nf the quine rosser pair has the same type as its projections and hence is termed a type level ordered pair hence this definition has the advantage of enabling a function defined as a set of ordered pairs to have a type only 1 higher than the type of its arguments this definition works only if the set of natural numbers is infinite this is the case in nf but not in type theory or in nfu j barkley rosser showed that the existence of such a type level ordered pair or even a type raising by 1 ordered pair implies the axiom of infinity for an extensive discussion of the ordered pair in the context of quinian set theories see holmes 1998 16 cantor frege definition edit early in the development of the set theory before paradoxes were discovered cantor followed frege by defining the ordered pair of two sets as the class of all relations that hold between these sets assuming that the notion of relation is primitive 17 x y r x r y displaystyle x y r xry this definition is inadmissible in most modern formalized set theories and is methodologically similar to defining the cardinal of a set as the class of all sets equipotent with the given set 18 morse definition edit morse kelley set theory makes free use of proper classes 19 morse defined the ordered pair so that its projections could 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 0 s x 1 s y displaystyle x y 0 times s x cup 1 times s y where the component cartesian products are kuratowski pairs of sets and where s x t t x displaystyle s x emptyset cup t mid t in x this renders possible pairs whose projections are proper classes the quine rosser definition above also admits proper classes as projections similarly the triple is defined as a 3 tuple as follows x y z 0 s x 1 s y 2 s z displaystyle x y z 0 times s x cup 1 times s y cup 2 times s z the use of the singleton set s x displaystyle s x which has an inserted empty set allows tuples to have the uniqueness property that if a is an n tuple and b is an m tuple and a b then n m ordered triples which are defined as ordered pairs do not have this property with respect to ordered pairs category theory edit commutative diagram for the set product x 1 x 2 a category theoretic product a b in a category of sets represents the set of ordered pairs with the first element coming from a and the second coming from b in this context the characteristic property above is a consequence of the universal property of the product and the fact that elements of a set x can be identified with morphisms from 1 a one element set to x while different objects may have the universal property they are all naturally isomorphic see also edit cartesian product abscissa and ordinate tarski grothendieck set theory trybulec andrzej 1989 tarski grothendieck set theory journal of formalized mathematics definition def5 of ordered pairs as x y x references edit lay steven r 2005 analysis with an introduction to proof 4th ed pearson prentice hall p 50 isbn 978 0 13 1...
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