Definition:Ordered Tuple as Ordered Set/Ordered Triple

Definition

The ordered triple $\tuple {a, b, c}$ of elements $a$, $b$ and $c$ can be defined either as the ordered pair:

$\tuple {a, \tuple {b, c} }$

or as the ordered pair:

$\tuple {\tuple {a, b}, c}$

where $\tuple {a, b}$ and $\tuple {b, c}$ are themselves ordered pairs.

Whichever definition is chosen does not matter much, as long as it is understood which is used. And even then, the importance is limited.

Also known as

An ordered triple is also sometimes seen named an ordered triad.

Some sources refer to it as just a triple, but it is usually important to stress that it is in fact ordered, so as not to confuse with a set with $3$ elements.

Some sources use the term $3$-tuple.

Also see

• Equality of Ordered Tuples

• Results about ordered triples can be found here.

Sources

• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.2$: Definition $1.3$
• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 3$. Ordered pairs; cartesian product 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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ordered pair
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ordered triple
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Ordered Pairs
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ordered pair
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ordered triple
• 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