# Homework - September 26, 2001

Topics Covered: Principle of Mathematical Induction, proofs by contradiction, vectors spaces (Sec. 2.1) Reading Assignment:
1. Appendix C - Fields pp. 510-513,
2. Handout
3. Sect. 1.2 and 1.3
Homework Problems:
1.
In no more than one page write a summary of what is induction and why it works as a method of proof. Assume your audience is someone with very little mathematical sophistication.
2.
Use induction to prove that

3.
Use induction to prove that if and is a positive integer.
4.
Find an error in the following inductive "proof" that all positive integers are equal. Let be the set of all such that equals all integers between 1 and . Then . Now suppose all integers up to and including are in . Then , so adding 1 to both sides gives that . Therefore, by the principle of mathematical induction, contains all positive integers, and so all positive integers are equal.
5.
Use the strong form of induction to prove that any integer can be expressed as , where and are nonnegative integers.
6.
Sect. 1.2 1,10,13,18, 22.

Math 24 Fall 2001 2001-09-24