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Contents
Contents
Discrete Mathematics
Dr. Th. Dana-Picard
Date:
June 10, 2002
dana@mail.jct.ac.il
©The author
Contents
Introduction and disclaimer
Elementary mathematical logic
Propositions
Quantifyers
Logical connectors
Logical equivalence
Negation
Inclusive or
Conjunction
Distributivity
De Morgan Laws
Tautologies and contradictions
Implication
Equivalence
Exclusive or
Sets
General settings
Operations with sets
Inclusion
Equality
Intersection
Union
Distributivity
Complementary set
Difference
Symmetric difference
The cartesian product of sets
Binary relations
The general case
Binary relations between elements of a set
Equivalence relations
Posets
Maximal and minimal elements.
Chains and antichains.
Products of posets
Mappings
Mappings
Injections
Surjections
Bijections
The characteristic function of a set
Permutations
Stirling numbers of the first kind
Transpositions and shuffles
Order of a permutation
Conjugacy classes in
The classical sets of numbers
The natural numbers
Addition
Multiplication
Ordering
Substraction
The integers
An equivalence relation on
The set of integers
Ordering
Division
The Euclidean Division
The Euclidean algorithm
Applications of the Euclidean algorithm
Rational numbers
Continued Fractions
Continued fractions for irrationals
Complexity of Euclidean Algorithm
Prime Numbers
Uniqueness of prime factorisation
Applications of prime factorisation
Modular Arithmetic
Solving Congruences
How to do it
Systems of congruences
Euler's Phi Function
Public Key Cryptography
Countability
Finite sets
Elementary combinatorics
Mappings from a finite set to a finite set
Arrangements
Permutations
Surjections
Combinations
Countable sets
Bigger sets
Induction and Counting
The Pigeonhole Principle
Induction
Strong Principle of Mathematical Induction
Recursive Definitions
Selection and Binomial Coefficients
Selections
Some more identities
Special Sequences of Integers
Stirling numbers of the first kind
Transpositions and shuffles
Stirling numbers of the second kind
Generating Functions
First method
Second Method
Catalan numbers
Bell numbers
Inclusion-Exclusion Principle
Introduction to graphs
Undirected graphs
Directed graphs
Multigraphs
Eulerian Digraphs
Bibliography
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