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 class of all sets:
A ~ A for all sets A. (IA: A A is a
Families of Sets and
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 not a definition.
In fact, technically, a family of sets
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 (PMI):
If S is a subset of such that;
1) 1 S, and
Nave Set Theory
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 however, a set will be thought
of as a collection of some
More Examples of 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 are an infinite number of primes.
Pf: BWOC suppose that
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 mathematics.
Alice saw a man with a telescope.
Two sisters were
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 premises and ending with the statement. However,
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 and B are disjoint finite sets with |A|=n
and |B| = m,
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 considered the "first" element and
the other the "second" ele
Completeness of the
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 interval centered at a with length 2.
So, N(5, .03) is th
Sample Final Exam
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 question).
Do your work on the blank sheets attached t
Completeness of the
Reals: a synopsis
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.
Likewise, l F is a lower bound for A iff l a for all
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 tell us anything
about the structure of a set.
An equivalence relation on a set S, is a relation on S which is
reflexive, symmetric and transitive.
Let S = and define R = cfw_(x,y) | x and y have the same parity
i.e., x and y are either both even or both odd.
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 ordered pairs, (i) and (ii) say that every element of A
Introduction to Abstract
I. General Class Organization
explain homework assignments
exams and quizzes
how to study mathematics
reading carefully is the key to success
you must participate to learn . mathematics is
NOT a spectator sport