Mathematics for computer scientists [electronic resource] : a practice-oriented approach / Peter Hartmann.

Intro
Preface
Contents
Part I Discrete Mathematics and Linear Algebra
1 Sets and Mappings
Abstract
1.1 Set Theory
Relationships between Sets
Operations with Sets
Calculation Rules for Set Operations
The Cartesian Product of Sets
1.2 Relations
Equivalence Relations
Order Relations
1.3 Mappings
The Cardinality of Sets
1.4 Comprehension Questions and Exercises
Anchor 14
2 Logic
Abstract
2.1 Propositions and Propositional Variables
Compound Propositions
Boolean Algebras
Evaluation of Propositional Formulas in a Program

2.2 Proof Principles
The Direct Proof
The Proof of Equivalence
The Proof by Contradiction
2.3 Predicate Logic (First-Order Logic)
Negation of Quantified Predicates
2.4 Logic and Testing of Programs
2.5 Comprehension Questions and Exercises
Anchor 15
3 Natural Numbers, Mathematical Induction, Recursion
Abstract
3.1 The Axioms of Natural Numbers
3.2 The Mathematical Induction
3.3 Recursive Functions
Recursions of Higher Order
Runtime Calculations for Recursive Algorithms
3.4 Comprehension Questions and Exercises
Anchor 9
4 Some Number Theory

Abstract
4.1 Combinatorics
4.2 Divisibility and Euclidean Algorithm
4.3 Modular Arithmetic
Calculating with Residue Classes
4.4 Hashing
Hash Functions
Collision Resolution
4.5 Comprehension Questions and Exercises
Anchor 11
5 Algebraic Structures
Abstract
5.1 Groups
Permutation groups
5.2 Rings
Polynomial Rings
5.3 Fields
The Field of Complex Numbers
The Field
5.4 Polynomial Division
Horner's Method
Residue Classes in the Polynomial Ring, The Field )
Using Polynomial Division for Error Detection
5.5 Elliptic Curves

Elliptic Curves over the Field of Real Numbers
Elliptic Curves Over Finite Fields
5.6 Homomorphisms
5.7 Cryptography
Encryption with Secret Keys
Encryption with Public Keys
The RSA Algorithm
The Diffie-Hellman Algorithm
The Diffie-Hellman Algorithm with Elliptic Curves
Key Generation
Random Numbers
5.8 Comprehension Questions and Exercises
Anchor 27
6 Vector Spaces
Abstract
6.1 The Vector Spaces , and
6.2 Vector Spaces
6.3 Linear Mappings
6.4 Linear Independence
6.5 Basis and Dimension of Vector Spaces

6.6 Coordinates and Linear Mappings
6.7 Comprehension Questions and Exercises
Anchor 10
7 Matrices
Abstract
7.1 Matrices and Linear Mappings in
Composition of linear mappings
7.2 Matrices and Linear Mappings from Kn ? Km
Matrix multiplication and composition of linear mappings
7.3 The Rank of a Matrix
7.4 Comprehension Questions and Exercises
Anchor 9
8 Gaussian Algorithm and Linear Equations
Abstract
8.1 The Gaussian Algorithm
8.2 Calculating the Inverse of a Matrix
8.3 Systems of Linear Equations