Math 213 - Homework 3
Assigned: 9/7/07
Due: 9/12/07 at the start of class.
Notation: Exercise a.b.c(d) stands for part (d) of Exercise c from Section a.b.
Problems:
(1) 3.3.4.
(2) 3.3.9.
(3) 3.3.26.
(4) Carefully explain why the running time of bubble sort is O(n2).
(5) 4.1.4.
(6) 4.1.10.
(7) 4.1.18.
(8) The Fibonacci sequence is defined by F0= 0, F1= 1, and Fn=Fn−1+Fn−2for n≥2.
Prove by induction that n
X
i=0
Fi=Fn+2 −1.
(9) Let A1, . . . , Ambe sets. Suppose that for any two sets Aiand Ajeither Ai⊆Ajor
Aj⊆Ai. Prove that there is some kso that Ai⊆Akfor all i. [Let P(n) be the
statement that there is a set Akwith 1 ≤k≤nso that Ai⊆Akfor 1 ≤k≤n.]
(10) Show that
n
X
i=1
1
i(i+ 2) =n(3n+ 5)
4(n+ 1)(n+ 2).
1