(Most chapters contains Exercises.)
List of Symbols.
1. A View of System Development.
Computing: a New Perspective.
General Problem Solving.
Computers and General Problem Solving.
Universal Features of Systems.
Modules: Components and Interfaces.
Establishment: Specification and Analysis.
System Development with Set Theory and Logic.
Introduction to Set Theory.
Induction and Recursion.
Formal Notation and Symbolic Logic.
The Practice of Model Establishment.
II. SET THEORY AND INDUCTION.
2. Sets and Basic Set Operations.
The Concept of Set.
Defining and Denoting Sets.
On the Caution Required in Defining Sets.
Three Fundamental Features of Sets.
Membership and Inclusion.
Special Sets and Set Cardinality.
Principle of Extensionality.
On the Method of Abstraction.
Basic Set Operations.
Disjointness and Related Operations.
Properties of Basic Set Operations.
Commutativity and Associativity.
Distributivity and DeMorgan's Laws.
Laws of the Empty Set, Idempotence and Absorption.
Laws of Inclusion.
Properties of Operations on Disjoint Sets.
Review of Notation.
Variables, Functions and Predicates.
Logical Connective Symbols.
Variants of Set Theory.
Russell's Paradox and Classes.
Set Theory as a Modelling Language.
Naive versus Axiomatic Set Theory.
3. Relations and Functions.
The Concept of a Relation.
Examples of Relations.
Some Derived Features of a Relation.
Basic Categories of Relations.
Class Relations AT.
The Concept of a Function.
Examples of Functions.
Basic Categories of Functions.
Fundamental Applications of Functions AT.
Characteristic Functions of Subsets and Relations.
Equivalence between Relations and Functions.
Families and Bags.
Families of Sets and Partitions.
n-ary Relations and Functions.
n-ary Cartesian Product.
n-ary Disjoint Union.
Examples of n-ary Relations.
Examples of n-ary Functions.
Nullary Functions and n-ary Operations.
n-ary Class Relations and Functions AT.
Operations on Relations and Functions AT.
Operations Returning Sets of Relations.
Composition of Relations and Functions.
Restrictions and Overriding.
Identity and Inclusion Functions.
Opposites and Inverses.
2.3 Induction and Recursion.
Deriving Peanos Postulates.
Significance of Peanos Postulates.
Inductive Sets with Respect to 0 and S.
A Concrete Definition of Natural NumbersAT.
Proof by Induction.
Proposition P(n); Example.
Proof by Induction: Theorem and Method.
Example of a Proof by Induction.
Alternative Application of the Induction Principle.
Second Induction Principle.
Example of a Simply Recursive Function.
Simple Recursion Theorem.
Variant of the Simple Recursion Theorem.
Defining Arithmetic Operations in NatAT.
4. Structural Induction.
General Definition of Structural Induction.
Proof by Structural Induction.
5. Free Inductive Sets; General Recursion Theorem.
Structural Recursion; Example.
General Recursion Theorem.
Second Variant of the General Recursion TheoremAT.
III. SYMBOLIC LOGIC.
3.1 Introduction to Symbolic Logic.
Two Features of Scientific Knowledge.
The Nature of Deductive Reasoning.
Purpose of Logic.
Logic versus Symbolic Logic.
Propositional Logic versus Predicate Logic.
3.2 Propositional Logic.
Informal Definition of Well-formed Formulas.
Formal Definition of Well-formed FormulasAT.
Semantics of Propositional Logic.
Recursive Definition of the Semantic Function M.
Well-definedness of M.
Definition of Tautological Implication.
Fundamental Tautologies and Applications.
3.3 First-order Predicate Logic.
Limitations of Propositional Logic.
Requirements to Be Met by First-order Logic.
Syntax of First-order Logic.
Formal Definition of Well-formed ExpressionsAT.
Unique Readability of Wfes.
Free and Bound Variables.
Semantics of First-order Logic.
Semantic Function of Well-formed Expressions.
Model and Satisfiability.
Definition of Logical Implication.
Examples of Valid Formulas.
3.4 Formal Deduction in First-order Logic.
Outline of a Formal Deduction System.
Generation of Formal Theorems.
Fundamental Properties of FDS.
Subset and Transitivity Properties of