# I've been struggling understanding Kuratowski's definition of ordered pairs. I understand what it means but I don't see why I should accept it. I've seen this question and this one, most importantly --- through reading the wiki page I've realised one thing. The only reason $(a,b)=\{\{a\},\{a,b\}\}$ is accepted is because it satisfies

The Activity: Ordered Pairs on the Coordinate Plane Activity. The Kuratowski definition isn't used because it captures some basic essence of ordered pair-ness

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}}. “is less than” among numbers as the set of ordered pairs (m, n) of nat- ural numbers   Which ordered pair is on the graph of the equation 2x+5y=4?? Reply. 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). The first number  I remember that ZFC, first-order Zermelo-Fraenkel set theory with the axiom of sets composed from the elements of A by repeated use of the pairing operation {x set-theoretic representation due to Kuratowski: [a, b] = {{a, 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 GOEDEL program does not assume Kuratowski's construction for ordered pairs, but this construction is nonetheless useful for deriving properties of cartesian products. In this notebook, the sethood rule for cartesian products is removed, and then rederived using the function KURA which maps ordered pairs to Kuratowski's model for them: Defining sets using pairs, check if definition satisfies the pair correctness property - Kuratowski ordered pair 1 Ordered pair operation (Kuratowski definition of) Yes, I disagree sustantively too. Definitions (e.g.

## 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,

{ { a }, { a, b } } The Wiener–Hausdorff–Kuratowski "ordered pair" definition 1914–1921. The history of the notion of "ordered pair" is not clear. As noted above, Frege (1879) proposed an intuitive ordering in his definition of a two-argument function Ψ(A, B). Pastebin.com is the number one paste tool since 2002. Pastebin is a website where you can store text online for a set period of time.

### 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.

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. Part of the problem is I haven't had a serious look at naive set theory since high school, but after reading the webs for a couple of hours, things are good for me except for this one piece.

There are several equivalent ways but since you mention Kuratowski, his definition is "The ordered pair, (a, b), is the set {a, {ab}}. That's closest to your (2) but does NOT mean "a is a subset of b". "a" and "b" theselves are not necessarily sets at all. I have found the following Kuratowski set definition of and ordered pair: (a,b) := {{a},{a,b}} Now I understand a set with the member a, and a set with the members a and b, but I am unsure how to read that, and how it describes an ordered pair, or Cartesian Coordinate. I would read the right side of that as "The set of sets {a} and {a,b}".
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As noted above, Frege (1879) proposed an intuitive ordering in his definition of a two-argument function Ψ(A, B). Pastebin.com is the number one paste tool since 2002. 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 $\zeta$.

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### 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). The first number

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### 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.

Or even a serious text on set theory may introduce an unordered pair as {a b}, where a b are the elements of the pair. Thus an unordered pair is simply a 1- or 2-element set.

## Kuratowski’s definition and Hausdorff's both do this, and so do many other definitions. 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. If we get that much, we are mathematically satisfied.

There are other definitions, of similar or lesser complexity, that are equally adequate: Kuratowski's definition of ordered pairs, (a, b)K := { {a}, {a, b}} is not clicking for me. Part of the problem is I haven't had a serious look at naive set theory since high school, but after reading the webs for a couple of hours, things are good for me except for this one piece. Consider an ordered pair which is (a,a).

the property desired of ordered pairs as stated above. 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. In 1921 Kazimierz Kuratowski offered the now-accepted definitioncf 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 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 .