An Introduction to Advanced Mathematics - Florida International University
An Introduction to Advanced Mathematics
M. Yotov
December 9, 2017
F or
...
Y ou
Know
1
W ho!
These Notes constitute a version of the course MAA 3200 Introduction to Advanced Mathematics
taught by the author at the Department of Mathematics and Statistics of FIU. The concepts of
classes, sets, relations, and functions are introduced and studied with rigour. Care is taken in motivating the introduction of the Zermelo-Fraenkel axioms. The natural numbers and the principle of
(finite) mathematical induction are discussed in detail. The integer, rational, and real numbers are
constructed and thoroughly discussed. As a brief introduction to Advanced Calculus, the classical
topology, convergent sequences of real numbers, and continuity of real valued function of a real
variable are studied. The Bolzano-Weierstrass Theorem, Intermediate Value Theorem, and Weierstrass¡¯s Theorem are proved.
Please send comments and corrections to the author at yotovm@fiu.edu .
c 2016 M.Yotov. Single paper copies for noncommercial personal use may be made without
explicit permission from the copyright holder.
2
Contents
1 Logic. Language of Proof
1.1 Propositions . . . . . . . . . . . . . . . .
1.2 Propositional Expressions, Tautologies .
1.3 Propositional Functions and Quantifiers
1.4 Methods of Proof . . . . . . . . . . . . .
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2 Sets
2.1 Undefined Terms, First Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Algebra of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Properties of N
3.1 The Least Element Principle (LEP) for N . .
3.2 The Principle of Math Induction . . . . . . .
3.3 Recursion, + and ¡¤ in N . . . . . . . . . . . .
3.4 What Are Natural Numbers; Peano¡¯s Axioms
3.5 Two Examples . . . . . . . . . . . . . . . . .
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4 Relations, Functions, and Orders
4.1 Class Relations and Class Functions
4.2 Relations from a Set to a Set . . . .
4.3 Functions from a Set to a Set . . . .
4.4 Pollency of Sets . . . . . . . . . . . .
4.5 Equivalence Relations . . . . . . . .
4.6 Orders . . . . . . . . . . . . . . . . .
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Systems
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5 Construction of the Standard Number
5.1 The Integers . . . . . . . . . . . . . . .
5.2 The Rational Numbers . . . . . . . . .
5.3 The Real Numbers . . . . . . . . . . .
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6 Topology on the Real Line
6.1 The Classical Topology on R . . . . . . . . . . . . .
6.2 Sequences of Real Numbers . . . . . . . . . . . . . .
6.3 Arithmetic Operations and a Relation on Sequences
6.4 Intro to Cantor¡¯s Real Numbers . . . . . . . . . . . .
7 The
7.1
7.2
7.3
7.4
Sets F(X, Y ), X, Y ? R
Limits of Functions . . . . . . . . . . . .
Compact Subsets of R . . . . . . . . . .
Compact Subsets of a Topological Space
Continuity of Functions . . . . . . . . .
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