8 1. The Real Numbers

16. Prove that a subset G of A × B is the graph of a function from A to B if and

only if the following condition is satisfied: for each a ∈ A there is exactly one

b ∈ B such that (a, b) ∈ G.

1.2. The Natural Numbers

The natural numbers are the numbers we use for counting, and so, naturally, they

are also called the counting numbers. They are the positive integers 1,2,3,....

The requirements for a system of numbers we can use for counting are very

simple. There should be a first number (the number 1), and for each number there

must always be a next number (a successor). After all, we don’t want to run out of

numbers when counting a large set of objects. This line of thought leads to Peano’s

axioms, which characterize the system of natural numbers N:

N1. There is an element 1 ∈ N.

N2. For each n ∈ N there is a successor element s(n) ∈ N.

N3. 1 is not the successor of an element of N.

N4. If two elements of N have the same successor, then they are equal.

N5. If a subset A of N contains 1 and is closed under succession (meaning s(n) ∈ A

whenever n ∈ A), then A = N.

Note: At this stage in the development of the natural number system, all

we have are Peano’s axioms; addition has not yet been defined. When we define

addition in N, S(n) will turn out to be n + 1.

Everything we need to know about the natural numbers can be deduced from

these axioms. That is, using only Peano’s axioms, one can define addition and

multiplication of natural numbers and prove that they have the usual arithmetic

properties. One can also define the order relation on the natural numbers and

prove that it has the appropriate properties. To do all of this is not diﬃcult, but

it is tedious and time consuming. We will do some of this here in the text and

the exercises, but we won’t do it all. We will do enough so that students should

understand how such a development would proceed. Then we will state and discuss

the important properties of the resulting system of natural numbers.

Our main tool in this section will be mathematical induction, a powerful tech-

nique that is a direct consequence of axiom N5.

Induction. Axiom N5 above is often called the induction axiom, since it is the

basis for mathematical induction. Mathematical induction is used in making defi-

nitions that involve a sequence of objects to be defined and in proving propositions

that involve a sequence of statements to be proved. Here, by a sequence we mean a

function whose domain is the natural numbers. Thus, a sequence of statements is

an assignment of a statement to each n ∈ N. For example, “n is either 1 or it is the

successor of some element of N” is a sequence of statements, one for each n ∈ N.

We will use induction to prove that all of these statements are true once we prove

the next theorem.