Homework #4
Due Thursday, September 17

1. The Well-Ordering Property of Natural Numbers states: ``Every nonempty subset S of the natural numbers has a smallest element.''
1. Use the Axiom of Induction to prove the Well-Ordering Property. Suggestion: Let S be any subset of the natural numbers . Assume that S has no smallest element. Let M be the complement of S. Prove that M equals and hence that S must be empty.
2. Use the Well-Ordering Property to prove the Axiom of Induction. Suggestion: Let M be any subset of satisfying I and II of the Axiom of Induction. Let S be the complement of M. Prove that S is empty and hence that M must be .
2. Suppose there are m people, m-1 black hats and m red hats. Assume individuals are given red hats and individuals are given black hats. Each person in order ( ) is asked in turn whether or not he/she can deduce the color of his/her hat. Prove by induction on n that person n can correctly deduce that his/her hat is red.
3. Induction problems #8, 10 (you can use the result of 9), 18.
4. Be sure you know how to do induction problems #2, 3, 4, 6, 9, 11, 12, but do not hand them in.

Carl Lee
Thu Sep 10 08:15:17 EDT 1998