Relations

Abstract

In colloquial usage we say that there is a “relation” between two things if there is some connection between them. An example of a relation between people is that of having the same color hair, and another example is that of one person being the child of another person. In mathematics we also discuss relations between objects, but, as is often the case, the technical meaning of the word “relation” in mathematics is not entirely the same as the colloquial use of the word. Some examples of relations between mathematical objects are very familiar, such as the relations = and < between real numbers.We saw some other relations in previous chapters, without having used the term “relation.” For example, we can define a relation between integers by saying that two integers a and b are related if and only if a|b. Relations (and especially equivalence relations, as discussed in Section 5.3), are used in crucial ways in many branches of mathematics, for example abstract algebra, number theory, topology and geometry.