Ordered pair

In mathematics, an ordered pair is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair is different from the ordered pair unless a = b. Ordered pairs are called 2-tuples, or sequences of length 2; the entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples. For example, the ordered triple can be defined. In the ordered pair, the object a is called the first entry, the object b the second entry of the pair. Alternatively, the objects are called the first and second components, the first and second coordinates, or the left and right projections of the ordered pair. Cartesian products and binary relations are defined in terms of ordered pairs. Let and be ordered pairs; the characteristic property of the ordered pair is: = if and only if a 1 = a 2 and b 1 = b 2. The set of all ordered pairs whose first entry is in some set A and whose second entry is in some set B is called the Cartesian product of A and B, written A × B.

A binary relation between sets A and B is a subset of A × B. The notation may be used for other purposes, most notably as denoting open intervals on the real number line. In such situations, the context will make it clear which meaning is intended. For additional clarification, the ordered pair may be denoted by the variant notation ⟨ a, b ⟩, but this notation has other uses; the left and right projection of a pair p is denoted by π1 and π2, or by πℓ and πr, respectively. In contexts where arbitrary n-tuples are considered, πni is a common notation for the i-th component of an n-tuple t. In some introductory mathematics textbooks an informal definition of ordered pair is given, such as For any two objects a and b, the ordered pair is a notation specifying the two objects a and b, in that order; this is followed by a comparison to a set of two elements. This "definition" is unsatisfactory because it is only descriptive and is based on an intuitive understanding of order. However, as is sometimes pointed out, no harm will come from relying on this description and everyone thinks of ordered pairs in this manner.

A more satisfactory approach is to observe that the characteristic property of ordered pairs given above is all, required to understand the role of ordered pairs in mathematics. Hence the ordered pair can be taken as a primitive notion, whose associated axiom is the characteristic property; this was the approach taken by the N. Bourbaki group in its Theory of Sets, published in 1954. However, this approach has its drawbacks as both the existence of ordered pairs and their characteristic property must be axiomatically assumed. Another way to rigorously deal with ordered pairs is to define them formally in the context of set theory; this can be done in several ways and has the advantage that existence and the characteristic property can be proven from the axioms that define the set theory. One of the most cited versions of this definition is due to Kuratowski and his definition was used in the second edition of Bourbaki's Theory of Sets, published in 1970; those mathematical textbooks that give an informal definition of ordered pairs will mention the formal definition of Kuratowski in an exercise.

If one agrees that set theory is an appealing foundation of mathematics all mathematical objects must be defined as sets of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set. Several set-theoretic definitions of the ordered pair are given below. Norbert Wiener proposed the first set theoretical definition of the ordered pair in 1914::=, he observed that this definition made it possible to define the types of Principia Mathematica as sets. Principia Mathematica had taken types, hence relations of all arities, as primitive. Wiener used instead of to make the definition compatible with type theory where all elements in a class must be of the same "type". With b nested within an additional set, its type is equal to


Prescote is a hamlet and civil parish about 4 miles north of Banbury in Oxfordshire. Its boundaries are the River Cherwell in the southeast, a tributary of the Cherwell called Highfurlong Brook in the west, Oxfordshire's boundary with Northamptonshire in the northeast. Prescote's toponym means "priest's cottage", referring to a cottage either owned by a priest or more inhabited by one. Legend associates Prescote with Saint Fremund, a Mercian prince held to have been martyred in the 9th century AD; the manor of Prescote is not listed in the Domesday Book of 1086, but had appeared by 1208-09, when the Bishop of Lincoln was the feudal overlord. Prescote comprised two manors that were held separately until 1417-1419, when John Danvers of Calthorpe, acquired both of them. In 1796 his descendant Sir Michael Danvers, 5th Baronet died without a male heir and left Prescote to his son-in-law Augustus Richard Butler. In 1798 Butler sold the estate to the Pares family, who in 1867 sold it to Samuel Jones-Loyd, 1st Baron Overstone.

In 1883 Baron Overstone died without a male heir and left his estates to his daughter, Lady Wantage. On her death in 1920 Prescote was sold to A. P. McDougall, whose Midland Marts company opened a cattle stockyard in 1921 beside Banbury Merton Street railway station. By 1964 Prescote belonged to Anne Crossman, the wife of Richard Crossman M. P. a descendant of the Danvers family. Prescote manor house has traces of a mediaeval moat, but a date-stone over the door of the present house indicates that it was built in 1691 by Sir Pope Danvers, 2nd Baronet; the house was extended early in the 19th century. The house at Prescote Manor Farm, about 0.5 miles northeast of the Manor House, is dated 1693. Prescote had a mill on the River Cherwell, called Boltysmylle in 1482 and Boltes Mill in 1613. By 1654 there was a "Prescote Mill". By 1703 the mill was in disrepair but its remains were still recorded as extant in 1797-98 and 1823. Today only its mill stream survives; the mill's decline may be linked with the manor's transition from arable to sheep farming.

In 1547 a Danvers leased land at Prescote to a shepherd, in 1797 it was reported that most of the 385 acres of the farm attached to Prescote Manor was "old inclosed" pasture. Crossley, Alan. H.. D. A.. F. A.. S.. A History of the County of Oxford, Volume 10: Banbury Hundred. Victoria County History. Pp. 206–210. ISBN 978-0-19-722728-2. CS1 maint: extra text: authors list Sherwood, Jennifer. Oxfordshire; the Buildings of England. Harmondsworth: Penguin Books. P. 560. ISBN 0-14-071045-0. Wass, Stephen. "Possible Early Christian Enclosure and Deserted Medieval Settlement at Prescote, Near Cropredy". Oxoniensia. Oxfordshire Architectural and Historical Society. LXXVI: 266–272. ISSN 0308-5562

Regional Internet registry

A regional Internet registry is an organization that manages the allocation and registration of Internet number resources within a region of the world. Internet number resources include IP autonomous system numbers; the regional Internet registry system evolved over time dividing the responsibility for management to a registry for each of five regions of the world. The regional Internet registries are informally liaised through the unincorporated Number Resource Organization, a coordinating body to act on matters of global importance; the African Network Information Center serves Africa. The American Registry for Internet Numbers serves Antarctica, parts of the Caribbean, the United States; the Asia-Pacific Network Information Centre serves East Asia, South Asia, Southeast Asia. The Latin America and Caribbean Network Information Centre serves most of the Caribbean and all of Latin America; the Réseaux IP Européens Network Coordination Centre serves Europe, Central Asia and West Asia. Regional Internet registries are components of the Internet Number Registry System, described in IETF RFC 7020.

The Internet Assigned Numbers Authority delegates Internet resources to the RIRs who, in turn, follow their regional policies to delegate resources to their customers, which include Internet service providers and end-user organizations. Collectively, the RIRs participate in the Number Resource Organization, formed as a body to represent their collective interests, undertake joint activities, coordinate their activities globally; the NRO has entered into an agreement with ICANN for the establishment of the Address Supporting Organisation, which undertakes coordination of global IP addressing policies within the ICANN framework. The Number Resource Organization is an unincorporated organization uniting the five RIRs, it came into existence on October 24, 2003, when the four existing RIRs entered into a memorandum of understanding in order to undertake joint activities, including joint technical projects and policy coordination. The youngest RIR, AFRINIC, joined in April 2005; the NRO's main objectives are to: Protect the unallocated IP number resource pool Promote and protect the bottom-up policy development process of the Internet Serve as a focal point for the Internet community to provide input on the RIR system A local Internet registry is an organization, allocated a block of IP addresses by a RIR, that assigns most parts of this block to its own customers.

Most LIRs are enterprises, or academic institutions. Membership in a regional Internet registry is required to become a LIR. Country code top-level domain Geolocation software Internet governance National Internet registry