Definition:Ordered Tuple as Ordered Set/Ordered Tuple
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Definition
The ordered tuple $\tuple {a_1, a_2, \ldots, a_n}$ of elements $a_1, a_2, \ldots, a_n$ is defined as either the ordered pair:
- $\tuple {a_1, \tuple {a_2, a_3, \ldots, a_n} }$
or as the ordered pair:
- $\tuple {\tuple {a_1, a_2, \ldots, a_{n - 1} }, a_n}$
where $\tuple {a_2, a_3, \ldots, a_n}$ and $\tuple {a_1, a_2, \ldots, a_{n - 1} }$ are themselves ordered tuples.
Whichever definition is chosen does not matter much, as long as it is understood which is used. And even then, the importance is limited.
Sources
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1964: A.M. Yaglom and I.M. Yaglom: Challenging Mathematical Problems With Elementary Solutions: Volume $\text { I }$ ... (previous) ... (next): Problems
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 5$: Products of Sets
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 8$: Cartesian product of sets
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.4$: Sets of Sets
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions
- 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Pairing
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Ordered Pairs
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 4$ The pairing axiom: $n$-tuples