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
1.7. Set Operations
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
3. REPRESENTING GRAPHS AND GRAPH ISOMORPHISM
3. Representing Graphs and Graph Isomorphism
3.1. Adjacency Matrix.
Definition 3.1.1. The adjacency matrix, A = [aij ], for a simple graph G =
(V, E ), where V = cfw_v1 , v2 , ., vn , is dened by
2. GRAPH TERMINOLOGY
2. Graph Terminology
2.1. Undirected Graphs.
Definitions 2.1.1. Suppose G = (V, E ) is an undirected graph.
(1) Two vertices u, v V are adjacent or neighbors if there is an edge e
between u and v .
The edge e connects u and v .
Introduction to Graph Theory
1. Introduction to Graphs
1.1. Simple Graphs.
Definition 1.1.1. A simple graph (V, E ) consists of a nonempty set representing vertices, V , and a set of unordered pairs of elements of V representing edges, E .
Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with
m rows and n columns is said to have dimension m n and may be represented as
a11 a12 a1n
a21 a22 a2n
. = [aij ]
3. APPLICATIONS OF NUMBER THEORY
3. Applications of Number Theory
3.1. Representation of Integers.
Theorem 3.1.1. Given an integer b > 1, every positive integer n can be expresses
n = ak bk + ak1 bk1 + + a1 b + a0 ,
where k 0, 0 a0 , a1 ,
2. INTEGERS AND ALGORITHMS
2. Integers and Algorithms
2.1. Euclidean Algorithm. Euclidean Algorithm. Suppose a and b are integers with a b > 0.
(1) Apply the division algorithm: a = bq + r, 0 r < b.
(2) Rename b as a and r as b and repeat until r = 0.
1. Integers and Division
Definition 1.1.1. Given two integers a and b we say a divides b if there is an
integer c such that b = ac. If a divides b, we write a|b. If a does not divide b, we
write a | b.
4. GROWTH OF FUNCTIONS
4. Growth of Functions
4.1. Growth of Functions. Given functions f and g , we wish to show how to
quantify the statement:
g grows as fast as f .
The growth of functions is directly related to the complexity of algorithms. We
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
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).
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
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).
2. METHODS OF PROOF
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.
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
A theorem is a proposit
3. PREDICATES AND QUANTIFIERS
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
2. PROPOSITIONAL EQUIVALENCES
2. Propositional Equivalences
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
1. Logic Denitions
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
2. Introduction to Functions
Definition 2.1.1. Let A and B be sets. A function
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
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
These are practice problems for the midterm. The midterm itself will consist
of 5 problems, from which you choose 4 to be graded.
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