9. Examples and Exercises

Two Examples:

The previous proof is "constructive" in that it actually provides us with an algorithm that allows us to "compute" from and Here are two simple examples:

1.

and and

Start with

Hence, by "induction"

Verify that this works. That is .

Now, define on , and on

2.

and and

Hence, again by "induction"

Verify that .

Exercise:

The Warmup.

, and

The Challenge:

and