# 2011-07-14 · 2: the concept of a pairing scheme, as constructed, depends on the concept of a mapping. Typically, a mapping is constructed as a set of ordered pairs (which can be encoded as Kuratowski sets). Plainly, there is something flawed about an argument that depends on Kuratowski pairs to assert the unimportance of Kuratowski pairs.

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 . In particular, it adequately expresses 'order', in that is false unless . There are other definitions, of similar or lesser complexity, that are equally adequate:

The currently accepted definition of an ordered pair was given by Kuratowski in 1921 (Enderton, 1977, pp. 36), though there exist several other definitions. Ordered pairs are also called 2-tuples, 2-dimensional vectors, or sequences of length 2. 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. An ordered pair is a pair of objects in which the order of the objects is significant and is used to distinguish the pair.

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An example is the ordered pair (a,b) which is notably different than the pair (b,a) unless the values of each variable are equivalent. Coordinates on a graph are represented by an ordered pair, x and y. Question: Use The Fundamental Property Of Ordered Pairs, But Not Kuratowski's Definition, To Show That If ((a, B), A) = (a, (b, A)), Then A = B. Use The Fundamental Property Of Ordered Pairs And Kuratowski's Definition To Show That A pair in which the components are ordered is basically an arrow between the components, which is sometimes called or analyzed as an interval within a larger context. Formalisations One may wish to declare ordered pairs to exist by fiat, which was done, for example, by both Bourbaki and Bill Lawvere . There are many mathematical definitions of ordered pair which have this property.

The Kuratowski construction allows this to be done withou The cartesian product of two sets needs to brought across from naive set theory into ZF set theory. Kuratowski's definition. In 1921 Kazimierz Kuratowski offered the now-accepted definitioncf introduction to Wiener's paper in van Heijenoort 1967:224.

## 2011-05-17

Known as: Pair (mathematics), Kuratowski ordered pair, Kuratowski pair. Expand. In mathematics, an ordered pair (a, b) is a pair of objects. The order in which 13 Feb 2017 Deﬁnition (Kuratowski) The ordered pair with coordinates x, y , denoted x, y , is the set {{x}, {x, y}} {x, y} tells that x and y are the components of Let x, y be objects.

### 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 $(a,b)\; =\; (x,y)\; \backslash leftrightarrow\; (a=x)\; \backslash land\; (b=y)$. In particular, it adequately expresses 'order', in that $(a,b)\; =\; (b,a)$ is false unless $b\; =\; a$.

Is (a,b) different from (a,a) when a=b? Next, what Cours netprof.fr de Mathématiques / DémonstrationProf : Jonathan It is an attempt to define ordered sets in terms of ordinary sets . We know that an n- tuple is different from the set of its coordinates. In an ordered set, the first element, second element, third element.. must be distinguished and identified. Definition of ordered pair in the Definitions.net dictionary.

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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 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 . In particular, it adequately expresses 'order', in that is false unless . There are other definitions, of similar or lesser complexity, that are equally adequate: 2: the concept of a pairing scheme, as constructed, depends on the concept of a mapping. Typically, a mapping is constructed as a set of ordered pairs (which can be encoded as Kuratowski sets). Plainly, there is something flawed about an argument that depends on Kuratowski pairs to assert the unimportance of Kuratowski pairs.

Is (a,b) different from (a,a) when a=b? Next, what
Cours netprof.fr de Mathématiques / DémonstrationProf : Jonathan
It is an attempt to define ordered sets in terms of ordinary sets . We know that an n- tuple is different from the set of its coordinates. In an ordered set, the first element, second element, third element..

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Information and translations of ordered pair in the most comprehensive dictionary definitions resource on the web. 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 (,) = (,) ↔ (=) ∧ (=). 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} . What is important is that the objects we choose to represent ordered pairs must behave like ordered pairs. If we get that much, we are mathematically satisfied. The Kuratowski definition isn't used because it captures some basic essence of ordered pair-ness, but because it does that we need it to do, which is just enough. using the function KURA which maps ordered pairs to Kuratowski's model for them: In[2]:= lambda pair x,y ,set set x ,set x,y Out[2]= KURA comment on notation The class set[x, y, ] is the class of all sets w such that w = x or w = y or .