3. RECURRENCE
120
3. Recurrence
3.1. Recursive Denitions. To construct a recursively dened function:
1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.
2. Recursion: Use a xed procedure (rule) to compute the value of the funct

2. PROPERTIES OF FUNCTIONS
111
2. Properties of Functions
2.1. Injections, Surjections, and Bijections.
Definition 2.1.1. Given f : A B
1. f is one-to-one (short hand is 1 1) or injective if preimages are unique. In
this case, (a = b) (f (a) = f (b).
2. f

CHAPTER 4
Applications of Methods of Proof
1. Set Operations
1.1. Set Operations. The set-theoretic operations, intersection, union, and
complementation, dened in Chapter 1.1 Introduction to Sets are analogous to the
operations , , and , respectively, tha

3. MATHEMATICAL INDUCTION
83
3. Mathematical Induction
3.1. First Principle of Mathematical Induction. Let P (n) be a predicate
with domain of discourse (over) the natural numbers N = cfw_0, 1, 2, .. If
(1) P (0), and
(2) P (n) P (n + 1)
then nP (n).
Term

2. METHODS OF PROOF
69
2. Methods of Proof
2.1. Types of Proofs. Suppose we wish to prove an implication p q . Here
are some strategies we have available to try.
Trivial Proof: If we know q is true then p q is true regardless of the
truth value of p.
Va

CHAPTER 3
Methods of Proofs
1. Logical Arguments and Formal Proofs
1.1. Basic Terminology.
An axiom is a statement that is given to be true.
A rule of inference is a logical rule that is used to deduce one statement
from others.
A theorem is a proposit

3. PREDICATES AND QUANTIFIERS
45
3. Predicates and Quantiers
3.1. Predicates and Quantiers.
Definition 3.1.1. A predicate or propositional function is a description of
the property (or properties) a variable or subject may have. A proposition may be
creat

2. PROPOSITIONAL EQUIVALENCES
33
2. Propositional Equivalences
2.1. Tautology/Contradiction/Contingency.
Definition 2.1.1. A tautology is a proposition that is always true.
Example 2.1.1. p p
Definition 2.1.2. A contradiction is a proposition that is alwa

CHAPTER 2
Logic
1. Logic Denitions
1.1. Propositions.
Definition 1.1.1. A proposition is a declarative sentence that is either
true (denoted either T or 1) or
false (denoted either F or 0).
Notation: Variables are used to represent propositions. The most

2. INTRODUCTION TO FUNCTIONS
15
2. Introduction to Functions
2.1. Function.
Definition 2.1.1. Let A and B be sets. A function
f: AB
is a rule which assigns to every element in A exactly one element in B .
If f assigns a A to the element b B , then we writ

CHAPTER 1
Introduction to Sets and Functions
1. Introduction to Sets
1.1. Basic Terminology. We begin with a refresher in the basics of set theory.
Our treatment will be an informal one rather than taking an axiomatic approach at
this time. Later in the s

MAA 4224Introduction to Analysis, Fall 2011
Practice Test
These are practice problems for the midterm. The midterm itself will consist
of 5 problems, from which you choose 4 to be graded.
1. Negate the following statements:
(a) For any a A and b B , we ha

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College Geometry MTG4212 Fall 2011
Homework 1
Due Wednesday, September 14, 2011
Write proofs of Propositions 9-12 (from section 1.2 of the text) carefully, making sure to
justify each step using postulates, common notions, and previously proved propositio

Understanding Cortical Folding Patterns in Development, Aging and Disease
Monica K. Hurdal
Department of Mathematics
Florida State University
Email: [email protected]
There is controversy and debate regarding the mechanisms involved in cortic

FALL 2011 SOPHOMORE HONORS RESEARCH COLLOQUIUM
HUM 2944-03
Wednesdays, 5:00- 6:15 p.m. in HCB 101
Dr. Alec Kercheval
[email protected]
Dr. Joe OShea
[email protected]
Course Description:
This is a 1 credit hour S/U course required for all Honors students wh

Useful Theorems on open and closed sets MAA 4224
The following is a list of denitions and theorem from class. To use this
list eectively, dont just read the list. See if you can state the denitions
without looking. Rewrite the proofs yourself, making sure

MAA4224 Introduction to Analysis Eriko Hironaka
Handout 2
Supplement to 1.2 Properties of sets
Notation: Notice that there is often an implied meta-set containing all given sets. Usually in this class the meta-set will be the set of real numbers. xA xA
c

MAC 2313, Section 04 with Dr. Hurdal
Fall 2011 Assignment 3
Due: Thursday November 10, 2011 at the beginning of class.
Please hand in complete solutions to the following problems. Points will be allocated for
clear and well written mathematical solutions.

MAC 2313, Section 04 with Dr. Hurdal
Fall 2011 Assignment 2
Due: Wednesday October 19, 2011 at the beginning of class.
With your group (3-4 people per group), please hand in complete written solutions (1 solution set per group in one hand writing) for the

MAC 2313, Section 04 with Dr. Hurdal
Fall 2011 Assignment 1
Due: Tuesday September 27, 2011 at the beginning of class.
With your group (3-4 people per group), please hand in complete written solutions (1 solution set per group in one hand writing) for the

Contents
Chapter 1. Introduction to Sets and Functions
1. Introduction to Sets
1.1. Basic Terminology
1.2. Notation for Describing a Set
1.3. Common Universal Sets
1.4. Complements and Subsets
1.5. Element v.s. Subsets
1.6. Cardinality
1.7. Set Operations

CHAPTER 7
Introduction to Relations
1. Relations and Their Properties
1.1. Denition of a Relation. Denition: A binary relation from a set A
to a set B is a subset
R A B.
If (a, b) R we say a is related to b by R.
A is the domain of R, and
B is the codomai