Section 8.3 Cartesian products of sets
Now when we were defining sets we used lists and the order in which we put things in the list didn’t matter. On some occasions we really need to put things into some order; we need a way to write an ordered pair of elements in which one of the pair is the first and the other is the second. Coordinates of points, in the plane are a very good example of this: the first number is the -coordinate (horizontal position) and the second is the -coordinate (vertical position), and we should not mix them up 99 . The point on the plane is not the same as the point We are used to this notation, “ ”, but we should define it before we go any further.
Given two sets the set of all possible ordered pairs is the Cartesian product of those sets. To be more precise:
Definition 8.3.2. Cartesian product.
Example 8.3.3.
Mind you, people rarely call the parts of an -coordinate by their correct names. The (ie the first of the pair) is called the abscissa and its use goes back at least as far as Fibonacci. The (the second of the pair) is the ordinate. These terms are not so common in modern English and people typically just call them -coordinate and -coordinate (which is a little jarring to the ear of the pedant).
While this is named for the French mathematician and philosopher Rene Descartes (1596 – 1650), it was also invented by Pierre de Fermat (1601 – 1665), and even earlier by Nicole Oresme (1325 – 1382). Fermat is famous for writing his “Last Theorem” in the margin of a book, and Oresme was the first to prove that the infinite series does not converge.