80-110   The Nature of Mathematical Reasoning

Summer 2000

Dirk Schlimm

Homework No. 3

Friday, July 7, 2000

Due Monday, July 10, 2000

1.  Write out the following two proofs, which were presented in class, as detailed and accurate as you can. State all your assumptions at the beginning and the conclusion at the end.  Justify every step you make. Also state the definitions that you are using. If you use variables, make sure to state what type they have, e.g., natural numbers, rational numbers.

  1. The square root of 2 is irrational.

  3. There are infinitely many prime numbers.

You may use the following lemmas without proving them:

2.  Determine the representation of 1978
  1. in base 60, and
  2. in base 2.