the rule is perfectly clear. I will often want to define functions by ``rules" of the following sort: is given by

It is not quite so clear that this is a rule, since the right side of (3.50) involves the function I am trying to define. However, if I try to use this rule to calculate , I get

and by this example, you recognize that (3.50) defines the familiar factorial function. In fact, I make this my definition of the factorial function.

We call the

and

but in , so I have , contradicting . So I see that (3.50) is

by

Thus,

We denote the value of by . Then we can rewrite (3.54) as

Note that and .

Proof: Define a proposition form on
by

Then which is true, since both sides of the equation are equal to . For all ,

and

Hence for all ,

By induction, is true for all , i.e.

The following results are easy to show and we will assume them.

I need this remark for the following definition to make sense.

Note that this definition of is consistent with our use of for multiplicative inverse. Also, this definition implies that

Proof: Let
, and write

then and and

It follows from theorem 3.60 that for all , and hence

Proof: By exercise 3.62

If we multiply both sides of this equation by , we get

i.e.