Cardinality
Equivalent Sets
Definition: We say that two sets are equivalent (sometimes called
equipotent), denoted by A ~ B iff there exists a bijection f: A B.
This is an equivalence relation on the
Families of Sets and
Extended Operations
Families of Sets
When dealing with sets whose elements are themselves sets it is
fairly common practice to refer to them as families of sets,
however this is n
Introduction to Abstract
Mathematics
I. General Class Organization
explain homework assignments
exams and quizzes
how to study mathematics
reading carefully is the key to success
you must participate
Math 3000
Introduction to Abstract Math
Homework
Chapter 2: 2.32(b,d)
Chapter 4: 4.2, 4.8, 4.14, 4.18
2.32 (b): If x2 1, then x 1 or x 1. In the latter case, however, Q(x) is false, so that the
entire
Math 3000
Introduction to Abstract Math
Solutions to Homework Collected 2/3/15
Due 2/3: (Reminder - you must explain all of your answers fully and in complete
sentences. An answer alone is not suffici
Math 3000
Introduction to Abstract Math
Solutions to Homework Collected 4/9/14
Homework
Chapter 8: 8.4, 8.12, 8.18, 8.22
8.4:
(a, 2) R1 and (2, a) R2 = (a, a) R
(a, 2) R1 and (2, b) R2 = (a, b) R
(a,
Math 3000
Introduction to Abstract Math
Solutions to Homework Collected 4/23/14
Problems that you must turn in
Chapter 6: 6.24
Chapter 8: 8.24, 8.26, 8.30, 8.71
6.24: Proof: Let x > 1. We proceed by i
Math 3000
Introduction to Abstract Math
Homework
Chapter 3: 3.53
Chapter 4: 4.42, 4.44,
Chapter 5: 5.26, 5.32, 5.34
3.53) Proof We proceed by cases based on the parity of x. Assume first that x is eve
Math 3000
Introduction to Abstract Math
Solutions to Homework Collected 4/2/14
Homework
Chapter 6: 6.10, 6.14, 6.30(a)
6.10)Proof:
Base Case: If n = 1, then a =
a(1r n )
1r ,
so the base case holds.
I
Math 3000
Introduction to Abstract Math
Solutions to Homework Collected 3/5/15
Homework
Chapter 4: 4.46, 4.54, 4.56, 4.78
4.46) Proof: We wish to prove the following:
() : If A B = A B, then A = B.
()
Math 3000
Introduction to Abstract Math
Due 2/19:
Homework
Chapter 2: 2.14
Chapter 3: 3.2, 3.6, 3.20, 3.58
Chapter 4: 4.24
2.14)
(a) At most one of my library books is overdue.
(b) Either both of my t
Math 3000
Introduction to Abstract Math
Solutions to Homework Collected 4/16/14
Problems that you must turn in
Chapter 6: 6.30(b)
Chapter 8: 8.28, 8.38, 8.64, 8.77, 8.78
6.30(b) Base Case: If m = 1, t
Mathematical Induction
The Induction Principle
In the axiomatic construction of the natural numbers = cfw_1, 2,
3, . (Peano, 1899), one of the axioms needed is the Principle of
Mathematical Induction
Nave Set Theory
Basic Definitions
Nave set theory is the non-axiomatic treatment of set theory.
In the axiomatic treatment, which we will only allude to at times,
a set is an undefined term. For us ho
Functions
Definition
A function f from A to B is a relation from A to B such that:
(i) Dom(f) = A, and
(ii) If (x,y) f and (x,z) f then y = z.
If A = B, we say that f is a function on A.
In terms of o
Equivalence Relations
Definition
An equivalence relation on a set S, is a relation on S which is
reflexive, symmetric and transitive.
Examples:
Let S = and define R = cfw_(x,y) | x and y have the same
The Reals
Outline
As we have seen, the set of real numbers, , has cardinality c.
This doesn't tell us very much about the reals, since there are many
sets with this cardinality and cardinality doesn't
Completeness of the
Reals: a synopsis
Bounds Again
Let A be a subset of an ordered field F. We say that
uF is an upper bound for A iff a u for all aA. If A
has an upper bound, A is bounded from above.
Math 3000
Sample Final Exam
Name _
Answer ten (10) of the following questions, you must choose four (4) questions from Part
A, the remaining six from either part. Exam totals 150 points (15 points per
Completeness of the
Reals
Neighborhoods
For any real number a, if is a positive real number, the
-neighborhood of a is the set
N(a, ) = cfw_x : | x - a | < = (a -, a +).
That is N(a, ) is the open in
Relations
Ordered Pairs
Given a non-empty set S, an ordered pair of elements of S, denoted
by (a, b), consists of a pair of elements of S ( a and b, which need
not be distinct) for which one is consid
Combinatorial Proofs
Two Counting Principles
Some proofs concerning finite sets involve counting the number
of elements of the sets, so we will look at the basics of counting.
Addition Principle: If A
Proof Methods
What is a proof?
Proofing as a social process, a communication art.
Theoretically, a proof of a mathematical statement is no
different than a logically valid argument starting with
some
III. Elementary Logic
The Language of Mathematics
While we use our natural language to transmit our mathematical
ideas, the language has some undesirable features which are not
acceptable in mathemati
More Examples of Proofs
Contradiction Proofs
Definition: A prime number is an integer greater than 1 which is divisible
only by 1 and itself. Ex: 2, 5, 11 are primes; 6, 15, 100 are not primes.
There
Math 3000
Introduction to Abstract Math
Solutions to Homework Collected 5/7/14
Due 5/7:
Homework
Chapter 9: 9.12, 9.24
9.12)
9.24) We claim that f (x) = x2 + 4x + 9 is neither one-to-one nor onto. Fir