Cover -- Preface -- Motivation -- Feature of the Book -- Acknowledgements -- Contents -- Chapter 1. Discrete Theory, Relations and Functions -- 1.1 Introduction -- 1.2 Elementary Theory of Sets -- 1.3 Set Rules and Sets Combinations -- 1.3.1 Rule of Equality -- 1.3.2 Study of Sets Combinations -- 1.3.3 Power Set -- 1.3.4 Multisets -- 1.3.5 Ordered Sets -- 1.3.6 Cartesian Products -- 1.4 Relations -- 1.4.1 Binary Relation -- 1.4.2 Equivalence Relation -- 1.4.3 Pictorial Representation of Relations -- 1.4.4 Composite Relation -- 1.4.5 Ordering Relation -- 1.5 Function -- 1.5.1 Classification of Functions -- 1.5.2 Composition of Functions -- 1.5.3 Inverse Functions -- 1.5.4 Recursively Defined Functions -- 1.6 Mathematical Induction and Piano's Axioms -- Chapter 2. Discrete Numeric Functions and Generating Functions -- 2.1 Introduction -- 2.2 Properties of Numeric Functions -- 2.2.1 Addition of numeric functions -- 2.2.2 Multiplication of Numeric Functions -- 2.2.3 Multiplication with a Scalar Factor to Numeric Function -- 2.2.4 Quotient of Numeric Functions -- 2.2.5 Modulus of Numeric Function -- 2.2.6. S1an and S1an -- 2.2.7 Accumulated Sum of Numeric Fuctions -- 2.2.8 Forward Difference & Backward Difference -- 2.2.9 Convolution of Numeric Functions -- 2.3 Asymptotic Behavior (Performance) of Numeric Functions -- 2.3.1 Big-Oh (O) Notation -- 2.3.2 Omega (Ω) Notation -- 2.3.3 Theta (θ) Notation -- 2.4 Generating Functions -- 2.5 Application of Generating Function to Solve Combinatorial Problems -- Chapter 3. Recurrence Relations with Constant Coefficients -- 3.1 Introduction -- 3.2 Recurrence Relation for Discrete Numeric Functions -- 3.3 Finding the Solution of LRRCC -- 3.3.1 Method of Finding Homogenous Solution -- 3.3.2 Method of Finding Particular Solution -- 3.4 Alternate Method (Finding Solution of LRRCC by Generating Function).

3.5 Common Recurrences from Algorithms -- 3.6 Method for Solving Recurrences -- 3.6.1 Iteration Method -- 3.6.2 Master Theorem -- 3.6.3 Substitution Method -- 3.7 Matrix Multiplication -- Chapter 4. Algebraic Structure -- 4.1 Introduction -- 4.2 Groups -- 4.3 Semi Subgroup -- 4.4 Complexes -- 4.5 Product Semigroups -- 4.6 Permutation Groups -- 4.7 Order of a Group -- 4.8 Subgroups -- 4.9 Cyclic Groups -- 4.10 Cosets -- 4.11 Group Mapping -- 4.12 Rings -- 4.13 Fields -- Chapter 5. Propositional Logic -- 5.1 Introduction to Logic -- 5.2 Symbolization of Statements -- 5.3 Equivalence of Formula -- 5.4 Propositional Logic -- 5.4.1 Well Formed Formula (wff) -- 5.4.2 Immediate Subformula -- 5.4.3 Subformula -- 5.4.4 Formation Tree of a Formula -- 5.4.5 Truth Table -- 5.5 Tautology -- 5.6 Theory of Inference -- 5.6.1 Validity by Truth Table -- 5.6.2 Natural Deduction Method -- 5.6.3 Analytical Tableaux Method (ATM) -- 5.7 Predicate Logic -- 5.7.1 Symbolization of Statements Using Predicate -- 5.7.2 Variables and Quantifiers -- 5.7.3 Free and Bound Variables -- 5.8 Inference Theory of Predicate Logic -- Chapter 6. Lattice Theory -- 6.1 Introduction -- 6.2 Partial Ordered Set -- 6.3 Representation of a POSET (Hasse Diagram) -- 6.4 Lattices -- 6.4.1 Properties of Lattices -- 6.4.2 Lattices and Algebraic Systems -- 6.4.3 Classes of Lattices -- 6.4.4 Product of Lattices -- 6.4.5 Lattice Homomorphism -- Chapter 7. Introduction to Languages and Finite Automata -- 7.1 Basic Concepts of Automata Theory -- 7.1.1 Alphabets -- 7.1.2 Strings -- 7.1.3 Power of Σ -- 7.1.4 Languages -- 7.2 Deterministic Finite State Automata (DFA) -- 7.2.1 Definition -- 7.2.2 Representation of a DFA -- 7.2.3 δ-head -- 7.2.4 Language of a DFA -- 7.3 Nondeterminstic Finite State Automata (NDFSA) -- 7.3.1 Definition -- 7.3.2 Representation -- 7.3.3 δ-head -- 7.3.4 Properties of δ-head.

7.3.5 Language of an NFA -- Chapter 8. Equivalence of NFA and DFA -- 8.1 Relationship Between NFA & DFA -- 8.2 Method of Conversion from NFA to DFA -- 8.3 Finite Automata with e moves -- 8.3.1 NFA with e moves -- 8.3.2 δ-head -- 8.3.3 Language of NFA with e-moves -- 8.3.4 Method of Conversion from NFA with e-moves to NFA -- 8.3.5 Equivalence of 'NFA with e-moves' to DFA -- Chapter 9. Regular Expressions -- 9.1 Introduction to Regular Expressions -- 9.2 Definition of Regular Expression -- 9.3 Equivalence of Regular Expression and Finite Automata -- 9.3.1 Construction of NFA with e-moves from Regular Expression -- 9.3.2 Construction of DFA from Regular Expression -- 9.4 Finite Automata to Regular Expression -- 9.4.1 Construction of DFA from Regular Expression -- 9.5 Construction of Regular Expression from DFA -- 9.6 Finite Automatons with Output -- 9.6.1 Melay Automaton -- 9.6.2 Moore Automaton -- 9.7 Equivalence of Melay & Moore Automatons -- 9.7.1 Equivalent Machine Construction (From Moore Machine-to-Melay Machine) -- 9.7.2 Melay Machine-to-Moore Machine -- Chapter 10. Regular and Nonregular Languages -- 10.1 Introduction -- 10.2 Pumping Lemma for Regular Languages -- 10.3 Regular and Nonregular Languages Solved Examples -- 10.4 Properties of Regular Languages -- 10.5 Decision Problems (DP) of Regular Languages -- 10.5.1 Emptiness Problem -- 10.5.2 Finiteness Problem -- 10.5.3 Membership Problem -- 10.5.4 Equivalence Problem -- 10.5.5 Minimization Problem and Myhill Nerode Theorem (Optimizing DFA) -- Chapter 11. Non-Regular Grammars -- 11.1 Introduction -- 11.2 Grammar -- 11.3 Classification of Grammar - Chomesky's Heirarchy -- 11.4 Sentential Form -- 11.5 Context Free Grammars (CFG) & Context Free Languages (CFL) -- 11.6 Derivation Tree (Parse Tree) -- 11.6.1 Parse Tree Construction -- 11.7 Ambiguous Grammar -- 11.7.1 Definition.

11.7.2 Ambiguous Context Free Language -- 11.8 Pushdown Automaton -- 11.9 Simplification of Grammars -- 11.10 Chomsky Normal Form (CNF) -- 11.11 Greibach Normal Form (GNF) -- 11.12 Pumping Lemma for Context Free Languages -- 11.13 Properties of Context Free Languages -- 11.14 Decision Problems of Context Free Languages -- 11.15 Undecided Problems of Context Free Languages -- Chapter 12. Introduction to Turing Machine -- 12.1 Introduction -- 12.2 Basic Features of a Turing Machine -- 12.2.1 Abstract View of a Turing Machine -- 12.2.2 Definition of a Turing Machine -- 12.2.3 Instantaneous Description of a Turing Machine -- 12.2.4 Representation of a Turing Machine -- 12.3 Language of a Turing Machine -- 12.4 General Problems of a Turing Machine -- 12.5 Turing Machine is the Computer of Natural Functions -- Appendix-Boolean Algebra -- A.1 Introduction -- A.2 Definition of Boolean Algebra -- A.3 Theorems of Boolean Algebra -- A.4 Boolean Functions -- A.5 Simplification of Boolean Functions -- A.6 Forms of Boolean Functions -- A.7 Simplification of Boolean Function Using K-Map.