Mathematical Induction
William Cherry
February 2011
These notes provide some additional examples to supplement the section of the text on mathematical induction.
Inequalities. It happens that often in mathematics, the more freedom one has in creating
a so
Math 3000.001 Homework 10
Solution
Do the following book problems:
Mance
1. 8.3
(a) f (x) = 2x + 1
(b) f (x) =
1
n+1
(c) f (x) = 1
(d) f (x) =
1
x
if x =
1
n+1
1
n
for some natural number n. Else f (x) = x.
if x = 1
1
n
for some natural number n. Else f
Math 3000.001 Homework 9
Due April 9th, 2013
Do the following book problems:
Mance
1. 7.5
(a) There are 3 3 = 9 functions, 3 2 = 6 injective functions and no surjective
functions.
(b) There are 2 2 2 2 = 8 functions, no injective functions, and 8 2 = 6
su
Math 3000.001 Homework 8
Due April 2nd, 2013
Do the following book problems:
Mance
1. 14.3
(a) F = cfw_(1/2, 3 1/n) : n N
(b) F = cfw_(n 1/2, n + 1/2) : n N
(c) F =
1
n
1
,1
3n2 n
1
3n2
: n N (see problem 6 on homework 1)
2. 14.6
(a) In = [n, ).
(b) In =
Math 3000.001 Homework 7
Due March 26th, 2013
Do the following book problems:
Mance
1. Read and understand theorem 12.11
2. Dont need to do 11.7 because it was on the last homework.
3. 12.3 (no justication required)
(a) 3, 3
(b) ,
(c) 4, 4
(d) 4, none
(e
Math 3000.001 Homework 5
Solutions
Mance
1. Do problem 10.15 in the book.
For the rst part, the base case was not proven. That is, P (n) needs to be
proven for n = 2 because the reasoning given only proves P (n + 1) from P (n)
if n 2. So P (1) isnt actual
Math 3000.001 Homework 4 Solutions
Mance
1. Prove that
3 is irrational.
Assume for contradiction that 3 is rational. So 3 = a/b, where a and b are
relatively prime integers. We may rewrite this as
a2 = 3b2 .
(1)
Thus, 3|a2 . We want to show that 3|a. We s
Math 3000.001 Homework 1 Solutions
Mance
1. Let A and B be subsets of a universal set U . Simplify each of the following
expressions.
(a) (A B ) (U \A) = B (A (U \A) = B U (by 5.13a)
=U
(b) (A B ) (U \A) = B (A (U \A) = B (by 5.13b)
=
(c) A (B (U \A) = (A