Search results
Results From The WOW.Com Content Network
The ordered pair (a, b) is different from the ordered pair (b, a), unless a = b. In contrast, the unordered pair, denoted {a, b}, always equals the unordered pair {b, a}. Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional ...
Kazimierz Kuratowski (Polish pronunciation: [kaˈʑimjɛʂ kuraˈtɔfskʲi]; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics .
By 1914 Norbert Wiener, using Whitehead and Russell's symbolism, eliminated axiom *12.11 (the "two-variable" (relational) version of the axiom of reducibility) by expressing a relation as an ordered pair using the null set. At approximately the same time, Hausdorff (1914, p. 32) gave the definition of the ordered pair (a, b) as {{a,1}, {b, 2
Kuratowski 1. Kazimierz Kuratowski 2. A Kuratowski ordered pair is a definition of an ordered pair using only set theoretical concepts, specifically, the ordered pair (a, b) is defined as the set {{a}, {a, b}}. 3. "Kuratowski-Zorn lemma" is an alternative name for Zorn's lemma Kurepa 1. Đuro Kurepa 2.
In NFU, these two definitions have a technical disadvantage: the Kuratowski ordered pair is two types higher than its projections, while the Wiener ordered pair is three types higher. It is common to postulate the existence of a type-level ordered pair (a pair (,) which is the same type as its projections) in NFU. It is convenient to use the ...
A 1‑tuple is called a single (or singleton), a 2‑tuple is called an ordered pair or couple, and a 3‑tuple is called a triple (or triplet). The number n can be any nonnegative integer . For example, a complex number can be represented as a 2‑tuple of reals, a quaternion can be represented as a 4‑tuple, an octonion can be represented as ...
'Ordered pairs = Kuratowski def., binary relations = sets of ordered pairs w/o explicit domain and codomain/range, functions = restricted binary relations, indexed families = syntactic sugar for functs., tuples = way to specify the members of the underlying set of an indexed family while allowing readers to assume that the indexing set is the ...
In 1914, Norbert Wiener showed how to code the ordered pair as a set of sets, making it possible to eliminate the relation types of Principia Mathematica in favor of the linear hierarchy of sets in TST. The usual definition of the ordered pair was first proposed by Kuratowski in 1921.