# Ordered Pair. more Two numbers written in a certain order. Usually written in parentheses like this: (12,5) Which can be used to show the position on a graph, where the "x" (horizontal) value is first, and the "y" (vertical) value is second. So (12,5) is 12 units along, and 5 units up.

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.

He was one of the leading representatives of the Warsaw School of Mathematics . $\begingroup$ 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 $2$. The concept of Kuratowski pair is one possible way of encoding the concept of an ordered pair in material set theory (say in the construction of Cartesian products ): A pair of the form. ( a, b) (a,b) is represented by the set of the form.

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. My points of confusion: 1. Hello. I have understood the Kuratowski definition of the ordered pair and appreciate it's usefulness but have a nagging difficulty about it. Consider an ordered pair which is (a,a). according to Kuratowski definition it is defined as {{a},{a,a}} . 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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Mathematical Structures Tuples are often used to encapsulate sets along with some operator or relation into a complete mathematical structure. Ordered pairs are also called 2-tuples, 2-dimensional vectors, or sequences of length 2.

### vectors in terms of sets, such a3 Kuratowski's device. We now discuss the ASL definition of ordered pair in terms of sets, and later will contrast it with other

36), though there exist several other An ordered pair a,b is not a set. It should be something The Kuratowski definition of an ordered pair is: a,b a , a,b .

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. My points of confusion: 1. Hello. I have understood the Kuratowski definition of the ordered pair and appreciate it's usefulness but have a nagging difficulty about it. Consider an ordered pair which is (a,a).

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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.

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.

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### 2012-10-20

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 property desired of ordered pairs as stated above.

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### Let x, y be objects. The Kuratowski ordered pair of x, y, with x being the first coordinate and y being the second coordinate, is defined to be the set

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## 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.

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 Kuratowski ordered pair of x, y, with x being the first coordinate and y being the second coordinate, is defined to be the set be expressed in set theory as a set of ordered pairs and since set theory provides a with these definitions is, though an ordered pair is defined to ess o r WO a pato. ولسم.

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.