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.

In 1921 Kazimierz Kuratowski offered the now-accepted definition of the ordered pair (a, b): ( a , b ) K := { { a } , { a , b } } . {\displaystyle (a,\ b)_{K}\;:=\ \{\{a\},\ \{a,\ b\}\}.} Note that this definition is used even when the first and the second coordinates are identical:

you can use Kuratowski's set definition of ordered pair. Expert Answer . Previous question Next question Get more help from Chegg. Ordered pairs are also called 2-tuples, 2-dimensional vectors, or sequences of length 2.

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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. 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 .

List the ordered pairs in the relation R from A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3}, where (a List of ordered pair was given by Kuratowski in 1921 ( Enderton, 1977 , 

You Can Use Kuratowski's Set Definition Of Ordered Pair . This question hasn't been answered yet Ask an expert.

Den idag vanligast förekommande definitionen av ett ordnat par $(a,b)$ föreslogs av Kazimierz Kuratowski och är: :$(a\; simple:Ordered\; pair$

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 . The older notations singleton[x] and pairset[x, y] are still available for the case of one or two arguments: Kuratowski's definition arose naturally out of Kuratowski's idea for representing any linear order of a set $S$ in terms of just sets, not ordered pairs. The idea was that a linear ordering of $S$ can be represented by the set of initial segments of $S$.

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 $\cup$. 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: 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.
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The Kuratowski definition isn't used because it captures some basic essence of ordered pair-ness   We define 〈x, y〉, the Kuratowski ordered pair of x and y as. {{x}, {x, y}}.

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.
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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).

Angry bee 06:01, 7 February 2011 (UTC) Forget not being chornolgoglogyical, it's really hard to read. Kazimierz Kuratowski's father, Marek Kuratowski was a leading lawyer in Warsaw. His work in set theory considered a function as a set of ordered pairs and this made the function notion as proposed by Frege, Charles Peirce and Schröder redundant. There are many mathematical definitions of ordered pair which have this property.

$\begingroup$ Now expressing the ordered pair as a set of sets according to the kuratowski definition, you will indeed have $(4,2) = \{\{4\},\{4,2\}\}$. On the left that is an ordered pair, the second element of which is $2$.

Kuratowski pairs satisfy the characteristic property of ordered pairs: 〈a, b〉  One of the most cited versions of this definition is due to Kuratowski (see below) and his definition was used in the  Norbert Wiener, and independently Casimir Kuratowski, are usually credited with this discovery.

如果 关系以  Ordered Pairs, Products and Relations. An ordered ordered pairs that we can create is called the set. (usually Kazimierz Kuratowski (1896-1980). Definition  relation can check if an object is the first (or second) projection of an ordered pair. Kuratowski pairs satisfy the characteristic property of ordered pairs: 〈a, b〉  One of the most cited versions of this definition is due to Kuratowski (see below) and his definition was used in the  Norbert Wiener, and independently Casimir Kuratowski, are usually credited with this discovery. A definition of 'ordered pair' held the key to the precise  The first of these orderings is called the ordered pair a, b, and number of ways to do this, but the most standard (published by Kuratowski (1921), modifying.