# Definition:Ordered Tuple/Ordered Couples and Ordered Pairs

## Definition

Notice the difference between ordered pairs and ordered couples.

By definition, an ordered couple $\left({a, b}\right)$ is in fact the set $\left\{{\left({1, a}\right), \left({2, b}\right)}\right\}$, where each of $\left({1, a}\right)$ and $\left({2, b}\right)$ 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 $\left({a, b}\right)$ 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): $\S 18$ - 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): $\S 6$