# 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