Definition:Ordered Pair/Informal Definition
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Definition
The definition of a set does not take any account of the order in which the elements are listed.
That is, $\set {a, b} = \set {b, a}$, and the elements $a$ and $b$ have the same status -- neither is distinguished above the other as being more "important".
An ordered pair is a two-element set together with an ordering.
In other words, one of the elements is distinguished above the other - it comes first.
Such a structure is written:
- $\tuple {a, b}$
and it means:
- first $a$, then $b$.
Coordinates
Let $\tuple {a, b}$ be an ordered pair.
The following terminology is used:
- $a$ is called the first coordinate
- $b$ is called the second coordinate.
This definition is compatible with the equivalent definition in the context of Cartesian coordinate systems.
Also see
Sources
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- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.2$
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.7$. Pairs. Product of sets
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Introduction: Set-Theoretic Notation
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 9$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.9$: Cartesian Product
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 5.2$
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 5$: Products of Sets
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 8$: Cartesian product of sets
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Ordered Pairs