chapter  12
10 Pages

Relations

As mentioned in the introductory remarks to Chapter 11, a matter of frequent interest in mathematical as well as in day to day experience is a comparison between objects a pair at a time. The two objects are selected, one from one set and one from perhaps another set. A test is applied or a comparison made between pairs of objects. The comparison is some relation, for example, "is the same age as," for persons or, "is less than," for real numbers. We want to distinguish when a given pair compares favorably with respect to the relation as opposed to not comparing favorably. Since we are dealing with pairs of things and since those two things might not be interchangeable, we need to use ordered pairs. For example, if P is a certain set of persons and N the set of natural numbers, we may be concerned with the set of pairs {(n,p)1(n,p)ENxP and n is the age of p}. This set of ordered pairs could be considered the age relation for the collection P, meaning that we include in the set, exactly those ordered pairs (n,p) for which the relation, "is the age of," applied to (n,p) is true. What we have is essentially an "age table" with the correct ages for the persons in P. For example, if Mary Jane Doe is 19 years old and she is in P, then (19, Mary Jane Doe) is in the age relation above. On the other hand, if John Doe is 23 years old, then (22, John Doe) is not in the age relation. If you are 20 and your name is M and you are in P, then (20, M) is in the age relation.