Relations and Equivalence Relations

In this section, we shall introduce a formal definition for the notion of a relation on a set. This is something we often take for granted in elementary algebra courses, but is a fundamental concept in mathematics i.e. the very notion of a function relies upon the definition of a relation. Following this, we shall discuss special types of relations on sets. 1. Binary Relations and Basic Definitions We start with a formal definition of a relation on a set S. Definition 1.1. A (binary) relation on a set S is a subset R of the Cartesian product S × S. If R is a relation and (x, y) ∈ R, then we say " x is related to y by R " or simply xRy. Example 1.2. The most familiar of all relations is the relation " = " (equals) which consists of all the elements (x, x) ∈ S × S. (less than with standard definition from the integers). Write down all the elements of R. (since it consists of all elements (x, y) with x < y). There are certain special properties a relation can have such as the following: Definition 1.4. Suppose R is a relation on a set S. Then we define the following: • We say R is reflexive if xRx for all x ∈ S • We say that R is symmetric if xRy implies yRx for all x, y ∈ S • We say R is transitive if xRy and yRz implies xRz for all x, y, z ∈ S We illustrate with some examples. Example 1.5. Show that the relation < (less than) on R is a transitive relation which is not symmetric or reflexive. Suppose x < y and y < z. Then clearly x < z and hence < is transitive. We do not have x < x and if x < y, then it is not the case that y < x, so it follows that it is neither reflexive or symmetric.