Ch 9: Injectivity, Surjectivity, Inverses & Functions on Sets
DEFINITIONS:
1. The identity function on a set X is the function
Suppose
for all
is a function. Then:
The image of f is defined to be:
The graph of f can be thought of as the set
We say that
.
Ralph (Eddie) Rise
Suk-Hyun Cho
Devin Kaylor
RC4 Encryption
RC4 is an encryption algorithm that was created by Ronald Rivest of RSA Security. It is used
in WEP and WPA, which are encryption protocols commonly used on wireless routers. The
workings of RC4
Graded Homework #1: Collected Wednesday 04/09
1. Which of the following expressions are statements? Circle the statements.
a) The US has 49 states.
b) I like to eat pizza and you hate going to the movies.
c) Call me on Thursday if you are home.
d) If we g
Math 310 A&BSpring 2008 Syllabus
Mathematics seems to endow one with something like a new sense.
Charles Darwin, in N. Rose (ed.) Mathematical Maxims and Minims.
Instructor: Alexandra Nichifor ([email protected]), office: Padelford C-326.
Class
Math 310: Review Problems for Final - Partial Solutions
1.
(i) Is L = 1/2 the limit of the sequence an =
denition of limit of a sequence.
n
2n+1 ?
Prove your answer using the
PARTIAL SOLUTION: Yes. Given an arbitrary > 0 you must nd an integer N
n
dependi
Math 310: Sample Problems for Homework 4
I. Review of formal DEFINITION of limit of a sequence:
Given a sequence of real numbers an , and a real number L, lim an = L i
n
> 0, N Z+ , such that n N, |an L| < .
This reads: For any positive real number epsil
Math 310: Collected Homework 6 (Ch. 12-13)
1. The set of all possible grades at Random University is G = cfw_4.0, 3.9, 3.8, ., 0.7 cfw_0.0.
Suppose there are 15 students enrolled in Math 903 at RU. At the end of the term, the
instructor must turn in a gra
Math 310: Homework 7 (Ch 14, 19, 20)
In pbls 2 and 3 below, you may assume that the Continuum Hypothesis holds.
PART 1 (Ch 14, do by 5/26):
1. Prove that the set of the irrational numbers is uncountable.
2. Determine the cardinality of the following sets
Math 310 Spring 2008: Proofs By Induction Worksheet Solutions
1. Prove that for all integers n 4, 3n n3 .
Scratch work:
(a) What is the predicate P(n) that we aim to prove for all n n0 ?
P (n) : 3n n3
(b) What is n0 =? = 4
(c) So the base case consists of
Math 310 Collected Homework #3: due Wed 4/23/2008
(Sets - plus a little induction and a few quantifiers)
1. Does the Trichotomy law hold for sets? In other words, given two sets A and B, is it true that
one of: A B, A B, or A=B must be true? If yes, give
Math 310, Homework 4 (Ch 7-9)
Collected Wednesday, April 30th
1. Problems II (page 117), problem 12.
2. Use the formal definition of limit of a sequence to prove that the sequence =
1
2
has limit equal to zero (that is, an is what the book calls a null se
Math 310: Collected Homework 5
1. (a) Let B be a proper subset of a set A. Prove that if there exists a
bijection f : A B, then the set A must be innite.
(b) Construct an explicit bijective function from the set of all integers Z to the set of natural num
Math 310: Review Problems for Final
Review the textbook and homework problems & proofs.
Review the midterm and midterm practice problems.
Additional practice problems:
1.
(i) Is L = 1/2 the limit of the sequence an =
the denition of limit of a sequence.
(
e Is Irrational
A. Bond, S. Honda, J. Nakamura, D. Pinkerton, S. Wu
May 28, 2008
1
Background
The mathematical constant e is very unique as it appears in many unexpected
places. e is used when continuously compounding bank accounts over time, it
is used i
Math 310 : Some Review Problems for Midterm 1
1. Prove that if A, B and C are sets such that C A and C B, then C A B
2. For all real numbers x and y, prove that |x + y|<|x| + |y|.
You may use the denition of |a|, the addition, multiplication and transitiv
Catalan Numbers
Written by Hung, Tzu-Hsiang(Ray)
May 30, 2008
Presented by Rachel Croft
In combinatorial mathematics, the Catalan numbers(1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012,
742900, 2674440, 9694845, .) form a sequence of natural
Pi is Irrational
By Jennifer, Luke, Dickson, and Quan
I.
Definition of Pi
II.
Proof of Lemma 2.5.1
III.
Proof that is irrational
IV.
Ivan Nivens Original Proof
Definition of
Pi is the Greek letter used in the formula to find the circumference, or perimet
Math 310: Proofs By Induction Worksheet
1. Prove that for all n 4, 3n n3 .
Scratch work:
(a) What is the predicate P(n) that we aim to prove for all n n0 ?
P (n) :
(b) What is n0 =? =
(c) So the base case consists of proving P (n0 ). Write out what this
m