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sions 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 148101 5 devlin keith 2004 sets functions and logic an introduction to abstract mathematics 3rd ed chapman hall crc p 79 isbn 978 1 58488 449 1 1 2 wolf robert s 1998 proof logic and conjecture the mathematician s toolbox w h freeman and co p 164 isbn 978 0 7167 3050 7 fletcher peter patty c wayne 1988 foundations of higher mathematics pws kent p 80 isbn 0 87150 164 3 quine has argued that the set theoretical implementations of the concept of the ordered pair is a paradigm for the clarification of philosophical ideas see word and object section 53 the general notion of such definitions or implementations are discussed in thomas forster reasoning about theoretical entities randall r dipert jun 1982 set theoretical representations of ordered pairs and their adequacy for the logic of relations canadian journal of philosophy 12 2 353 374 doi 10 1080 00455091 1982 10715803 jstor 40231262 wiener s paper a simplification of the logic of relations is reprinted together with a valuable commentary on pages 224ff in van heijenoort jean 1967 from frege to gödel a source book in mathematical logic 1979 1931 harvard university press cambridge ma isbn 0 674 32449 8 pbk van heijenoort states the simplification this way by giving a definition of the ordered pair of two elements in terms of class operations the note reduced the theory of relations to that of classes cf introduction to wiener s paper in van heijenoort 1967 224 cf introduction to wiener s paper in van heijenoort 1967 224 van heijenoort observes that the resulting set that represents the ordered pair has a type higher by 2 than the elements when they are of the same type he offers references that show how under certain circumstances the type can be reduced to 1 or 0 kuratowski casimir 1921 sur la notion de l ordre dans la théorie des ensembles fundamenta mathematicae 2 1 161 171 doi 10 4064 fm 2 1 161 171 this differs from hausdorff s definition in not requiring the two elements 0 and 1 to be distinct from a and b tourlakis george 2003 lectures in logic and set theory vol 2 set theory cambridge univ press proposition iii 10 1 for a formal metamath proof of the adequacy of short see here opthreg also see tourlakis 2003 proposition iii 10 1 j barkley rosser 1953 logic for mathematicians mcgraw hill holmes m randall on ordered pairs on boise state march 29 2009 the author uses σ 1 displaystyle sigma _ 1 for φ displaystyle varphi and σ 2 displaystyle sigma _ 2 for ψ displaystyle psi holmes m randall 1998 elementary set theory with a universal set archived 2011 04 11 at the wayback machine academia bruylant the publisher has graciously consented to permit diffusion of this monograph via the web frege gottlob 1893 144 grundgesetze der arithmetik pdf jena verlag hermann pohle archived from the original pdf on 2016 10 21 retrieved 2017 09 14 kanamori akihiro 2007 set theory from cantor to cohen pdf elsevier bv p 22 footnote 59 morse anthony p 1965 a theory of sets academic press v t e mathematical logic general axiom list cardinality first order logic formal proof formal semantics foundations of mathematics information theory lemma logical consequence model theorem theory type theory theorems list paradoxes gödel s completeness incompleteness theorems tarski s undefinability banach tarski paradox cantor s theorem paradox diagonal argument compactness halting problem lindström s löwenheim skolem russell s para...
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