I will assign problems for class discussion every day. You should do the listed problems before the corresponding class so that you are prepared to discuss them.
Periodically I will select a subset of the problems I have assigned to be handed in. The selected problems will be announced with a relatively short lead time. Thus it is in your interest to stay current on the homework.
Chapter 1 (page 20ff)
Chapter 2
Chapter 3
Chapter 4
Chapter 5 and The Book Of Proof
Additional Problems:
Prove that if \(p\) is prime then \(\binom{p}{r}\) is divisible by \(p\). Use this to prove that \(n^p\equiv n(\mod p)\) for all \(n\) and any prime number \(p\).
Let \(f(x)=(1+x)^{n}\). Prove that the \(i^{th}\) derivative \[ f^{i}(x)=\frac{n!}{(n-i)!}(1+x)^{n-i} \] and therefore \(f^{i}(0)=n!/(n-i)!\). Taylor’s formula says that \[ (1+x)^{n}=\sum_{i=0}^{n}\frac{f^{i}(0)}{i!}x^{i} \] while the binomial theorem says that \[ (1+x)^{n}=\sum_{i=0}^{n}\binom{n}{i}x^{i}. \] Use these facts to conclude that \(\binom{n}{i}=\frac{n!}{(n-i)!i!}\)
Oct 30: GV Chapter 5: 31, 37, 41, 42, 44, 47, 48.
Chapters 12 and 14 of The Book of Proof