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.

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

To understand what Kuratowski's school years were like it is necessary to look a little at the history of Poland around the time he was born. The first thing to note is that really Poland did not formally exist at this time. 7 Jul 2007 ture on the evolution of the Wiener-Kuratowski ordered pair, and a discussion by Quine of the merits of an ordered-pair implemen- tation that 26 Nov 2014 The standard definition of ordered pairs in set theory is credited to Kuratowski. By this definition, ( a , b ) is simply {{ a }, { a , b }}.

For example, the ordered triple (a,b,c) can be defined as (a, (b,c)), i.e., as one pair nested in another. Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2; ordered pairs of scalars are also called 2-dimensional vectors. In classical Euclidean geometry (that is in synthetic geometry), vectors were introduced (during 19th century) as equivalence classes, under equipollence, of ordered pairs of points; two pairs Therefore [latex]x = u[/latex] and [latex]y = v[/latex]. This property is useful in the formal definition of an ordered pair, which is stated here but not explored in-depth. The currently accepted definition of an ordered pair was given by Kuratowski in 1921 (Enderton, 1977, pp. 36), though there exist several other definitions. Kuratowski allows us to both work with ordered pairs and work in a world where everything is a set.

An ordered pair contains the coordinates of one point in the coordinate system. A point is named by its ordered pair of the form of (x, y).

Yes, I disagree sustantively too. Definitions (e.g. Kuratowski's definition) of ordered pair are restricted to pairs of sets, which are mathematical objects. There are also definitions of ordered pairs of classes, but that does not matter in this case, since classes are mathematical objects too. Ladislav Mecir 14:17, 15 September 2016 (UTC)

One way might be to use the Kuratowski encoding of ordered pairs, and use union as before, as well as a singleton-forming operation $\zeta$. We would therefore add to the STLC $\zeta$ and $\cup$.

Where T is the set of natural numbers, let Pair be the bijection: T×T 6 T described by the Kuratowski defined ordered pairs by Kuratowski = {{a,b},{a}}.

Previous question Next question Get more help from Chegg. 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. 2012-10-20 2.7 Ordered pairs 1. Introduction to set theory and to methodology and philosophy of mathematics and computer programming Ordered pairs An overview by Jan Plaza c 2017 Jan Plaza Use under the Creative Commons Attribution 4.0 International License Version of February 14, 2017 An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element.An ordered pair with first element a and second element b is usually written as (a, b).

The currently accepted definition of an ordered pair was given by Kuratowski in 1921 (Enderton, 1977, pp. 36), though there exist several other definitions. Kuratowski allows us to both work with ordered pairs and work in a world where everything is a set. While "custom-types" makes the everiday mathematical work easier, the set-theoretical "monoculture" makes the foundation comfortably more trust-worthy. It was the Polish mathematician Kazimierz Kuratowski who in 1921 came up with the definition that is now most commonly used: the one in which the ordered pair (a,b) is defined as the set {{a},{a,b}}. This definition, like the alternatives, has no deeper meaning other than that one can prove that the above property holds for it.
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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} .

Known as: Pair (mathematics), Kuratowski ordered pair, Kuratowski pair. Expand.
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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.

Which definition we pick is not really important. What is important is that the objects we choose to represent ordered pairs must behave like ordered pairs.

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浏览句子中ordered pair的翻译示例，听发音并学习语法。 In 1921, Kazimierz Kuratowski proposed a simplification of Wiener's definition of ordered pairs, and

An ordered pair is a pair of objects in which the order of the objects is significant and is used to distinguish the pair. 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… Ordered Pairs, Products and Relations An ordered pair is is built from two objects Ð+ß,Ñ ß+ ,Þand As the name suggests, the “order” matters: and are two different ordereÐ+ß,Ñ Ð,ß+Ñ +œ,Ñd pairs (unless .