Math form Words
Chapter Three - Order


James Adrian
      Many of our experiences with the idea of order require quantitative comparison. Used pencils can be arranged in order of length. We can use a ruler, or if the differences are visually obvious, we can make judgements of more than and less than. In either case, we are making quantitative comparisons. Distance, weight, time, brightness, and a great many other quantities can be ordered; but finding a way to define order for mathematical purposes cannot involve the use of any skill in judging quantities. We need to know what the meaning of order is before we define ways to compare quantities. This can't be allowed to be circular. Counting and calculating depend on the concept of order. A definition of order cannot depend on counting and calculating.

      The current task related to order is to find a way to define ordered collections and ordered sets. The path taken on this journey involves the concept of association.

      There are two cases to consider. Here is case 1: The people of a particular village are accustomed to frequently seeing Guinevere and Reginald together in public places. Guinevere is associated with Reginald, and Reginald is associated with Guinevere. Here is case 2: The children of a school class have learned to effortlessly recite the English alphabet from A to Z, but when asked to recite this alphabet backwards, they all find it difficult to do. Saying L, M, N, O, P is easy. Saying P, O, N, M, L, is a whole other thing to practice. This capability does not come automatically as a result of memorizing the alphabet in the usual way.

      Associations that go only in one direction are revery common. The Black Sea may remind you of water. The Black Sea is surely associated with water; but water will not reliably remind you of the Black Sea. Water is far from uniquely associated with the Black Sea. The association is not symmetrical. It is asymmetrical.

      Remembering the alphabet is remembering a list of asymmetrical associations. Remembering the alphabet backwards is remembering another list of asymmetrical associations.

      This is the notation for the symmetrical association:   G ←→ R

      This is the notation for an asymmetrical association:   D → E

      Since English is read from left to right, the right arrow, rather than the left, is chosen for the asymmetrical association.

      Definition - A symmetrical association, G ←→ R, associates R with G and associates G with R.

      Definition - An asymmetrical association, X → Y, associates Y with X, but does not associate X with Y.

      Definition - S is an ordered set if and only if each of the following statements is true:

      S is a set.

      S contains at least a pair of elements.

      S contains a single element, h, that is not associated with any other element in S.

      Each element in S, other than h, is associated with exactly a single other element in S.

      This is the notation for an ordered set, S, containing elements p, q, and r: S = (p, q, r)

      Definition - An unordered set, U, is a set, and U is not an ordered set.