MAT 300 Spring 04
Dr. Zandieh
Homework #8
Due April 8, 2004
Part I: Induction Proofs
a) Use the method of induction to prove 122 1
0
−= +
=
∑n
n
k
k
b) Note that if n 3, then .
Use this and induction to prove 2
≥1223
2+>+=≥⋅= nnnnnnn n > n2 for all n ≥5.
Part II: Recall the problem from class with the dial and clicks.
Define [1] to be the set of all integers, Z, which are equivalent to 1 (mod 6).
I.e. [1] = { | More generally define [y] = Z∈x(mod6)}.1≡x}(mod6)|{ yxx ≡
∈
Z.
Prove that the family F = {[1], [2],[3],[4],[5],[6]} is a partition of the integers.
Part III: Let be the equivalence relation on Z given by: m≡
≡
n
⇔
(m and n are the same
distance from 8 on the number line).
a) Completely describe the partition of Z given by
≡
.
b) Prove that your answer to a) is a partition of Z.