Transcript PowerPoint-1

Introduction to Real
Analysis
Dr. Weihu Hong
Clayton State University
8/19/2008
Review of Real Number
System
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R denotes the set of real numbers;
Q denotes the set of rational numbers;
(R, +, •) forms a field;
A field is a set F with two operations, addition (+) and
multiplication (•), that satisfy the following axioms:
 F is closed under both operations;
 Both operations are commutative;
 Both operations are associative;
 There exist additive identity 0 and multiplicative identity 1;
 Every element has an additive inverse;
 Every nonzero element has a multiplicative inverse
 Multiplication is distributive over addition.
Is (Q, +, •) a field?
Order Properties
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P denotes the set of positive real numbers.
The following are two order properties:
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(O1) If a and b belong to P, then so do a + b and a•b.
(O2) If a belongs to R, then one and only one of the
following hold:
a  P,a  P, a  0.
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Any field F with a nonempty subset satisfying the order
properties is called an ordered field. Therefore, (R, +, •)
is an ordered field.
Is (Q, +, •) an ordered field?
Is (C, +, •) an ordered field? Where C denotes the set of
complex numbers.
The meanings of Inequality
signs
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Let a and b belong to R.
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If a – b belongs to P, then we write a > b or b < a;
If a – b belongs to P or a – b = 0, then we write a b.
Basic results:
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Upper Bound of a set
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Definition of upper bound of a set: A subset E of R is
bounded above if there exists β in R such that for each x in
E, x β. Such a β is called an upper bound of E.
How should you define “bounded below” and “lower bound”?
A set E is bounded if E is bounded both above and below.
Examples:
 A = {0, ½, 2/3, ¾, …}. Is A bounded below? Is A bounded
above? Is A bounded?
 N = {1, 2, 3, …}. Is N bounded below? Is N bounded
above? Is N bounded?
 B = {r belongs to Q: 0 < r and r²<2}. Is B bounded below?
Is B bounded above? Is B bounded? Does B have the
maximum element?