Introduction

Suppose that we are using the notations of mathematical logic, as described in Chapter 2. We have some universe of discourse E, and let us suppose that a predicate R(x) is given. We could look at the objects in E in turn, and check for each object whether or not R(x) is true. If we then selected the collection of objects which made R(x) true, we would have what is known as a set.

Although the predicate calculus allows us to do anything we desire, it has been found over the last century or so that set notation is far more convenient and natural. A predicate in one variable is thought of as expressing a property of objects in the universe of discourse; a set is the collection of all those objects with that property. Conversely, given a collection of objects in the universe of discourse, we can form a predicate which expresses the property of being in that collection. Sets and one place predicates are completely interchangeable.