Introduction Edit. In mathematics, an ordered pair is a collection of two objects, where one of the objects is first (the first coordinate or left projection), and the other is second (the second coordinate or right projection).

In mathematics, an ordered pair (a, b) is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. (In contrast, the unordered pair {a, b} equals the unordered pair {b, a}.) In the ordered pair (a, b), the object a is called the first entry, and the object b the second entry of the pair.

In particular, it adequately expresses 'order', in that is false unless . There are other definitions, of similar or lesser complexity, that are equally adequate: 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. Illustrated definition of Ordered Pair: Two numbers written in a certain order. Usually written in parentheses like this: (12,5) Which Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of notation since an ordered pair need not be an element of a vector 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).

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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  av G Hamrin · 2005 · Citerat av 11 — That is, D is the set of ideals of (P,⊑), ordered by the inclusion of iterated limits, satisfies the following generalisation [5] of the Kuratowski. av J Eklund · 2016 · Citerat av 4 — a function f : A → B is a binary relation, i.e. a set of ordered pairs (a, b), such According to Kuratowski's theorem (Bondy & Murty, 2008, p. 268) a graph is  couple, couplet, distich, duad, duet, duo, dyad, ordered pair, pair, span, twain, 1.1 Kuratowskis definition; 1.2 Wieners definition; 1.3 Hausdorffs definition.

Intuitively, for Kuratowski's definition, the first element of the ordered pair, X, is a member of all the members of the set; the second element, Y, is the member not common to all the members of the set - if there is one, otherwise, the second element is identical to the first element. The idea Kuratowski's definition, the first element of the ordered pair, X, is a member of all the members of the set; the second element, Y, is the member not common to all the members of the set - if there is one, Kuratowski's definition.

Kazimierz Kuratowski's father, Marek Kuratowski was a leading lawyer in Warsaw. 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.

(In contrast, the unordered pair {a, b} equals the unordered pair {b, a}.) In the ordered pair (a, b), the object a is called the first entry, and the object b the second entry of the pair. $\begingroup$ This is a situation where categorical thinking is really helpful: you should define "ordered pairs" by a universal property, run the usual argument to show that if they exist then they are unique up to a canonical isomorphism, and then use any construction you want to actually show that they exist. You then only use the universal property when you prove results about them, so Kuratowski finiteness is defined as follows. Given any set S, the binary operation of union endows the powerset P(S) with the structure of a semilattice.

12 Jun 2017 The currently accepted definition of an ordered pair was given by Kuratowski in 1921 (Enderton, 1977, pp. 36), though there exist several other 

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].

Now consider an ordered triplet (a,a,a) it 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 -tuple is defined inductively using the construction of an ordered pair.
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Pastebin is a website where you can store text online for a set period of time. Ordered Pairs, Products and Relations An ordered pair is is built from two objects Ð+ß,Ñ ß+ ,Þand As the name suggests, Kazimierz Kuratowski (1896-1980). However, suppose we wanted to do this sort of iterative process in the STLC with ordered pairs, forming $(g, b)$ and then $(a, g, b)$. One way might be to use the Kuratowski encoding of ordered pairs, and use union as before, as well as a singleton-forming operation $\zeta$.

The Kuratowski definition you quoted doesn't mention the terms "first member of the ordered pair " and "second member of the ordered pair", so it's fair to say the Kuratowski definition tells us nothing about the meaning of those terms.
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Den idag vanligast förekommande definitionen av ett ordnat par $(a,b)$ föreslogs av Kazimierz Kuratowski och är: :$(a\; simple:Ordered\; pair$

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. Hey all, I have a very basic question. Kuratowski's definition of ordered pairs, (a, b)K := {{a}, {a, b}} is not clicking for me.