Sets, Logic and Maths for Computing

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Sets, Logic and Maths for Computing / David Makinson

資料種別:: 電子ブック
出版情報:: Springer eBooks Computer Science
Springer London, 2008
著者名:: Makinson, David
ISBN:: 9781846288449 [1846288444]
9786613437884 [6613437883]
書誌ID:: OB00350741

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言語:

英語

目次情報:

Collecting Things Together: Sets / 1：

The Intuitive Concept of a Set / 1.1：

Basic Relations between Sets / 1.2：

Inclusion / 1.2.1：

Identity / 1.2.2：

Proper Inclusion / 1.2.3：

Euler Diagrams / 1.2.4：

Venn Diagrams / 1.2.5：

Ways of Defining a Set / 1.2.6：

The Empty Set / 1.3：

Emptiness / 1.3.1：

Disjoint Sets / 1.3.2：

Boolean Operations on Sets / 1.4：

Intersection / 1.4.1：

Union / 1.4.2：

Difference and Complement / 1.4.3：

Generalised Union and Intersection / 1.5：

Power Sets / 1.6：

Some Important Sets of Numbers / 1.7：

Comparing Things: Relations / 2：

Ordered Tuples, Cartesian Products, Relations / 2.1：

Ordered Tuples / 2.1.1：

Cartesian Products / 2.1.2：

Relations / 2.1.3：

Tables and Digraphs for Relations / 2.2：

Tables for Relations / 2.2.1：

Digraphs for Relations / 2.2.2：

Operations on Relations / 2.3：

Converse / 2.3.1：

Join of Relations / 2.3.2：

Composition of Relations / 2.3.3：

Image / 2.3.4：

Reflexivity and Transitivity / 2.4：

Reflexivity / 2.4.1：

Transitivity / 2.4.2：

Equivalence Relations and Partitions / 2.5：

Symmetry / 2.5.1：

Equivalence Relations / 2.5.2：

Partitions / 2.5.3：

The Correspondence between Partitions and Equivalence Relations / 2.5.4：

Relations for Ordering / 2.6：

Partial Order / 2.6.1：

Linear Orderings / 2.6.2：

Strict Orderings / 2.6.3：

Closing with Relations / 2.7：

Transitive Closure of a Relation / 2.7.1：

Closure of a Set under a Relation / 2.7.2：

Associating One Item with Another: Functions / 3：

What is a Function? / 3.1：

Operations on Functions / 3.2：

Domain and Range / 3.2.1：

Image, Restriction, Closure / 3.2.2：

Composition / 3.2.3：

Inverse / 3.2.4：

Injections, Surjections, Bijections / 3.3：

Injectivity / 3.3.1：

Surjectivity / 3.3.2：

Bijective Functions / 3.3.3：

Using Functions to Compare Size / 3.4：

The Equinumerosity Principle / 3.4.1：

The Principle of Comparison / 3.4.2：

The Pigeonhole Principle / 3.4.3：

Some Handy Functions / 3.5：

Identity Functions / 3.5.1：

Constant Functions / 3.5.2：

Projection Functions / 3.5.3：

Characteristic Functions / 3.5.4：

Families of Sets / 3.5.5：

Sequences / 3.5.6：

Recycling Outputs as Inputs: Induction and Recursion / 4：

What are Induction and Recursion? / 4.1：

Proof by Simple Induction on the Positive Integers / 4.2：

An Example / 4.2.1：

The Principle behind the Example / 4.2.2：

Definition by Simple Recursion on the Natural Numbers / 4.3：

Evaluating Functions Defined by Recursion / 4.4：

Cumulative Induction and Recursion / 4.5：

Recursive Definitions Reaching Back more than One Unit / 4.5.1：

Proof by Cumulative Induction / 4.5.2：

Simultaneous Recursion and Induction / 4.5.3：

Structural Recursion and Induction / 4.6：

Defining Sets by Structural Recursion / 4.6.1：

Proof by Structural Induction / 4.6.2：

Defining Functions by Structural Recursion on their Domains / 4.6.3：

Condition for Defining a Function by Structural Recursion / 4.6.4：

When the Unique Decomposition Condition Fails? / 4.6.5：

Recursion and Induction on Well-Founded Sets / 4.7：

Well-Founded Sets / 4.7.1：

The Principle of Proof by Well-Founded Induction / 4.7.2：

Definition of a Function by Well-Founded Recursion on its Domain / 4.7.3：

Recursive Programs / 4.8：

Counting Things: Combinatorics / 5：

Two Basic Principles: Addition and Multiplication / 5.1：

Using the Two Basic Principles Together / 5.2：

Four Ways of Selecting k Items out of n / 5.3：

Counting Formulae: Permutations and Combinations / 5.4：

The Formula for Permutations (O+R-) / 5.4.1：

The Formula for Combinations (O-R-) / 5.4.2：

Counting Formulae: Perms and Coms with Repetition / 5.5：

The Formula for Permutations with Repetition Allowed (O+R+) / 5.5.1：

The Formula for Combinations with Repetition Allowed (O-R+) / 5.5.2：

Rearrangements and Partitions / 5.6：

Rearrangements / 5.6.1：

Counting Partitions with a Given Numerical Configuration / 5.6.2：

Weighing the Odds: Probability / 6：

Finite Probability Spaces / 6.1：

Basic Definitions / 6.1.1：

Properties of Probability Functions / 6.1.2：

Philosophy and Applications / 6.2：

Some Simple Problems / 6.3：

Conditional Probability / 6.4：

Interlude: Simpson's Paradox / 6.5：

Independence / 6.6：

Bayes' Theorem / 6.7：

Random Variables and Expected Values / 6.8：

Random Variables / 6.8.1：

Expectation / 6.8.2：

Induced Probability Distributions / 6.8.3：

Expectation Expressed using Induced Probability Functions / 6.8.4：

Squirrel Math: Trees / 7：

My First Tree / 7.1：

Rooted Trees / 7.2：

Labelled Trees / 7.3：

Interlude: Parenthesis-Free Notation / 7.4：

Binary Search Trees / 7.5：

Unrooted Trees / 7.6：

Definition of Unrooted Tree / 7.6.1：

Properties of Unrooted Trees / 7.6.2：

Finding Spanning Trees / 7.6.3：

Yea and Nay: Propositional Logic / 8：

What is Logic? / 8.1：

Structural Features of Consequence / 8.2：

Truth-Functional Connectives / 8.3：

Tautologicality / 8.4：

The Language of Propositional Logic / 8.4.1：

Assignments and Valuations / 8.4.2：

Tautological Implication / 8.4.3：

Tautological Equivalence / 8.4.4：

Tautologies and Contradictions / 8.4.5：

Normal Forms, Least letter-Sets, Greatest Modularity / 8.5：

Disjunctive Normal Form / 8.5.1：

Conjunctive Normal Form / 8.5.2：

Eliminating Redundant Letters / 8.5.3：

Most Modular Representation / 8.5.4：

Semantic Decomposition Trees / 8.6：

Natural Deduction / 8.7：

Enchainment / 8.7.1：

Second-Level (alias Indirect) Inference / 8.7.2：

Something about Everything: Quantificational Logic / 9：

The Language of Quantifiers / 9.1：

Some Examples / 9.1.1：

Systematic Presentation of the Language / 9.1.2：

Freedom and Bondage / 9.1.3：

Some Basic Logical Equivalences / 9.2：

Semantics for Quantificational Logic / 9.3：

Interpretations / 9.3.1：

Valuating Terms under an Interpretation / 9.3.2：

Valuating Formulae under an Interpretation: Basis / 9.3.3：

Valuating Formulae under an Interpretation: Recursion Step / 9.3.4：

The x-Variant Reading of the Quantifiers / 9.3.5：

The Substitutional Reading of the Quantifiers / 9.3.6：

Logical Consequence etc / 9.4：

Natural Deduction with Quantifiers / 9.5：

Index

Collecting Things Together: Sets / 1：

The Intuitive Concept of a Set / 1.1：

Basic Relations between Sets / 1.2：

Inclusion / 1.2.1：

Identity / 1.2.2：

Proper Inclusion / 1.2.3：

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