Definition:Ordered Tuple/Ordered Couples and Ordered Pairs
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Definition
Notice the difference between ordered pairs and ordered couples.
By definition, an ordered couple $\tuple {a, b}$ is in fact the set $\set {\tuple {1, a}, \tuple {2, b} }$, where each of $\tuple {1, a}$ and $\tuple {2, b}$ are ordered pairs.
It is not possible to use the definition of ordered couple as the definition of ordered pair, as the latter is used to define a mapping, which is then used to define an ordered couple.
However, in view of the Equality of Ordered Tuples, it is generally accepted that it is valid to use the notation $\tuple {a, b}$ to mean both an ordered couple and an ordered pair.
It is worth bearing this in mind, as there are times when it is important not to confuse them.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.4$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.15$: Sequences
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations