Modern Analysis, Homework 2
Due June 10, 2010
1.
(2 pts) Suppose
f
:
A
→
B
is a function. Prove that
f
is a bijection if and only if it is
both injective (onetoone) and surjective (onto).
2.
(2 pts) Prove by induction that for all positive integers
n
,
n
X
i
=1
i
=
n
(
n
+ 1)
2
.
3.
(5 pts) Write down an explicit bijection
N
×
N
→
N
.
Hint:
count by “diagonals” and use
problem
2.
4.
(5 pts) Rudin, Ch 2, problem 2. If we haven’t talked about complex numbers in class,
just read about them in Rudin. You may also assume the following result from algebra: let
P
(
x
) be a polynomial of degree
d
. Then
P
has at most
d
roots (this follows from the division
algorithm). (
Just for fun:
The set of algebraic integers is actually a field!!)
5.
(1 pt) A complex number is called
transcendental
if it is not the root of any polynomial
with integral coefficients. Show that the set of transcendentals is uncountable.
6.
Define a
positive integer sequence of positive integers
(abbr: PISPI) to be just a function
f
:
N
→
N
. You may also use the sequence notation,
{
a
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 Spring '11
 Peters
 Integers, pts, 1 pt, 3 pts, 5 pts, 2 pts

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