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Vector (mathematics and physics)

vectorvectorsvectorial
Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2; ordered pairs of scalars are also called 2-dimensional vectors.
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

tuplesn-tuple5-tuple
Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2; ordered pairs of scalars are also called 2-dimensional vectors.
-tuple is defined inductively using the construction of an ordered pair.

Cartesian product

productCartesian squareCartesian power
Cartesian products and binary relations (and hence functions) are defined in terms of ordered pairs. 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, and written A × B.
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A×B, is the set of all ordered pairs (a, b) where a is in A and b is in B.

Function (mathematics)

functionfunctionsmathematical function
Cartesian products and binary relations (and hence functions) are defined in terms of ordered pairs.
A function is uniquely represented by the set of all pairs

Unordered pair

binary alphabetBinary set
(In contrast, the unordered pair {a, b} equals the unordered pair {b, a}.)
In contrast, an ordered pair (a, b) has a as its first element and b as its second element.

Binary relation

relationrelationsidentity relation
Cartesian products and binary relations (and hence functions) are defined in terms of ordered pairs. A binary relation between sets A and B is a subset of A × B.
In mathematics, a binary relation over two sets A and B is a set of ordered pairs (a, b) consisting of elements a of A and elements b of B.

Norbert Wiener

WienerWiener, NorbertN. Wiener
Norbert Wiener proposed the first set theoretical definition of the ordered pair in 1914:
In that dissertation, he was the first to state publicly that ordered pairs can be defined in terms of elementary set theory.

Set theory

axiomatic set theoryset-theoreticset
If one agrees that set theory is an appealing foundation of mathematics, then all mathematical objects must be defined as sets of some sort.
, is the set whose members are all possible ordered pairs

Set (mathematics)

setsetsmathematical set
If one agrees that set theory is an appealing foundation of mathematics, then all mathematical objects must be defined as sets of some sort. 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, and written A × B.
The Cartesian product of two sets A and B, denoted by A × B is the set of all ordered pairs (a, b) such that a is a member of A and b is a member of B.

Principia Mathematica

Principiaramified theory of typesRamified type theory
:He observed that this definition made it possible to define the types of Principia Mathematica as sets.
"Relations" are what is known in contemporary set theory as sets of ordered pairs.

New Foundations

NFUNFTyped set theory
In type theory and in outgrowths thereof such as the axiomatic set theory NF, the Quine–Rosser pair has the same type as its projections and hence is termed a "type-level" ordered pair.
In 1914, Norbert Wiener showed how to code the ordered pair as a set of sets, making it possible to eliminate relation types in favor of the linear hierarchy of sets described here.

Bracket

parentheses{brackets
The definition short is so-called because it requires two rather than three pairs of braces.
In set theory, chevrons or parentheses are used to denote ordered pairs and other tuples, whereas curly brackets are used for unordered sets.

Axiom of regularity

axiom of foundationFoundationfoundation axiom
Proving that short satisfies the characteristic property requires the Zermelo–Fraenkel set theory axiom of regularity.
See ordered pair for specifics.

Kazimierz Kuratowski

KuratowskiCasimir KuratowskiKuratowski, Kazimierz
In 1921 Kazimierz Kuratowski offered the now-accepted definition

Morse–Kelley set theory

Kelley–Morse set theoryKMMK
Morse–Kelley set theory makes free use of proper classes.
Pairing licenses the unordered pair in terms of which the ordered pair, may be defined in the usual way, as.

Product (category theory)

productproductscategorical product
A category-theoretic product A × B in a category of sets represents the set of ordered pairs, with the first element coming from A and the second coming from B.
the ordered pair

Tarski–Grothendieck set theory

Axiom of UniversesTarski's axiomTarski-Grothendieck set theory
Also note that the short version is used in Tarski–Grothendieck set theory, upon which the Mizar system is founded.)

Mathematics

mathematicalmathmathematician
In mathematics, an ordered pair (a, b) is a pair of objects.

Sequence

sequencessequentialinfinite sequence
Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2; ordered pairs of scalars are also called 2-dimensional vectors.

Scalar (mathematics)

scalarscalarsbase field
Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2; ordered pairs of scalars are also called 2-dimensional vectors.

Subset

supersetproper subsetsubsets
A binary relation between sets A and B is a subset of A × B.

Interval (mathematics)

intervalopen intervalclosed interval
notation may be used for other purposes, most notably as denoting open intervals on the real number line.

Real line

real number linereal axisline
notation may be used for other purposes, most notably as denoting open intervals on the real number line.

Primitive notion

primitivePrimitivesnotion
Hence the ordered pair can be taken as a primitive notion, whose associated axiom is the characteristic property.

Nicolas Bourbaki

BourbakiBourbaki groupBourbaki, Nicolas
This was the approach taken by the N. Bourbaki group in its Theory of Sets, published in 1954.