2 1. The Real Numbers

devoted to using these properties to develop rigorous proofs of the main theorems

of calculus.

1.1. Sets and Functions

We precede our study of the real numbers with a brief introduction to sets and

functions and their properties. This will give us the opportunity to introduce the

set theory notation and terminology that will be used throughout the text.

Sets. A set is a collection of objects. These objects are called the elements of

the set. If x is an element of the set A, then we will also say that x belongs to A or

x is in A. A shorthand notation for this statement that we will use extensively is

x ∈ A.

Two sets A and B are the same set if they have the same elements – that is, if every

element of A is also an element of B and every element of B is also an element of

A. In this case, we write A = B.

One way to define a set is to simply list its elements. For example, the statement

A = {1, 2,3,4}

defines a set A which has as elements the integers from 1 to 4.

Another way to define a set is to begin with a known set A and define a new

set B to be all elements x ∈ A that satisfy a certain condition Q(x). The condition

Q(x) is a statement about the element x which may be true for some values of x

and false for others. We will denote the set defined by this condition as follows:

B = {x ∈ A : Q(x)}.

This is mathematical shorthand for the statement “B is the set of all x in A such

that Q(x)”. For example, if A is the set of all students in this class, then we might

define a set B to be the set of all students in this class who are sophomores. In this

case, Q(x) is the statement “x is a sophomore”. The set B is then defined by

B = {x ∈ A : x is a sophomore}.

Example 1.1.1. Describe the set (0,3) of all real numbers greater than 0 and less

than 3 using set notation.

Solution: In this case the statement Q(x) is the statement “0 x 3”. Thus,

(0,3) = {x ∈ R : 0 x 3}.

A set B is a subset of a set A if B consists of some of the elements of A – that

is, if each element of B is also an element of A. In this case, we use the shorthand

notation

B ⊂ A.

Of course, A is a subset of itself. We say B is a proper subset of A if B ⊂ A and

B = A.

For example, the open interval (0,3) of the preceding example is a proper subset

of the set R of real numbers. It is also a proper subset of the half-open interval

(0,3] – that is, (0,3) ⊂ (0,3], but the two are not equal because the second contains

3 and the first does not.