Definition:Ordered Tuple/Ordered Couples and Ordered Pairs

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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.