A Bridge to Higher Mathematics
Textbooks in Mathematics Series

A Bridge to Higher Mathematics is more than simply another book to aid the transition to advanced mathematics. The authors intend to assist students in developing a deeper understanding of mathematics and mathematical thought.

The only way to understand mathematics is by doing mathematics. The reader will learn the language of axioms and theorems and will write convincing and cogent proofs using quantifiers. Students will solve many puzzles and encounter some mysteries and challenging problems.

The emphasis is on proof. To progress towards mathematical maturity, it is necessary to be trained in two aspects: the ability to read and understand a proof and the ability to write a proof.

The journey begins with elements of logic and techniques of proof, then with elementary set theory, relations and functions. Peano axioms for positive integers and for natural numbers follow, in particular mathematical and other forms of induction. Next is the construction of integers including some elementary number theory. The notions of finite and infinite sets, cardinality of counting techniques and combinatorics illustrate more techniques of proof.

For more advanced readers, the text concludes with sets of rational numbers, the set of reals and the set of complex numbers. Topics, like Zorn?s lemma and the axiom of choice are included. More challenging problems are marked with a star.

All these materials are optional, depending on the instructor and the goals of the course.

Elements of logic

True and false statements

Logical connectives and truth tables

Logical equivalence

Quantifiers

Proofs: Structures and strategies

Axioms, theorems and proofs

Direct proof

Contrapositive proof

Proof by equivalent statements

Proof by cases

Existence proofs

Proof by counterexample

Proof by mathematical induction

Elementary Theory of Sets. Functions

Axioms for set theory

Inclusion of sets

Union and intersection of sets

Complement, difference and symmetric difference of sets

Ordered pairs and the Cartersian product

Functions

Definition and examples of functions

Direct image, inverse image

Restriction and extension of a function

One-to-one and onto functions

Composition and inverse functions

*Family of sets and the axiom of choice

Relations

General relations and operations with relations

Equivalence relations and equivalence classes

Order relations

*More on ordered sets and Zorn's lemma

Axiomatic theory of positive integers

Peano axioms and addition

The natural order relation and subtraction

Multiplication and divisibility

Natural numbers

Other forms of induction

Elementary number theory

Aboslute value and divisibility of integers

Greatest common divisor and least common multiple

Integers in base 10 and divisibility tests

Cardinality. Finite sets, infinite sets

Equipotent sets

Finite and infinite sets

Countable and uncountable sets

Counting techniques and combinatorics

Counting principles

Pigeonhole principle and parity

Permutations and combinations

Recursive sequences and recurrence relations

The construction of integers and rationals

Definition of integers and operations

Order relation on integers

Definition of rationals, operations and order

Decimal representation of rational numbers

The construction of real and complex numbers

The Dedekind cuts approach

The Cauchy sequences approach

Decimal representation of real numbers

Algebraic and transcendental numbers

Comples numbers

The trigonometric form of a complex number