Relation mathematics: Encyclopedia II - Relation mathematics - Example: divisibility

Relation mathematics - Example: divisibility

A more typical example of a 2-place relation in mathematics is the relation of divisibility between two positive integers n and m that is expressed in statements like "n divides m" or "n goes into m". This is a relation that comes up so often that a special symbol "|" is reserved to express it, allowing one to write "n|m" for "n divides m".

To express the binary relation of divisibility in terms of sets, we have the set P of positive integers, P = {1, 2, 3, …}, and we have the binary relation D on P such that the ordered pair (n, m) is in the relation D just in case n|m. In other turns of phrase that are frequently used, one says that the number n is related by D to the number m just in case n is a factor of m, that is, just in case n divides m with no remainder. The relation D, regarded as a set of ordered pairs, consists of all pairs of numbers (n, m) such that n divides m.

For example, 2 is a factor of 4, and 6 is a factor of 72, which two facts can be written either as 2|4 and 6|72 or as D(2, 4) and D(6, 72).

Adapted from the Wikipedia article "Example: divisibility", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki