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Discrete mathematics

Author: R K Bisht; H S Dhami
Publisher: New Delhi, India : Oxford University Press, 2015.
Series: Oxford higher education.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"A textbook designed for the students of computer science engineering, information technology, and computer applications to help them develop the foundation of theoretical computer science ... Adopting a solved problems approach to explain the concepts, the book presents numerous theorems, proofs, practice exercises, and multiple-choice questions."--Page 4 of cover.
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Genre/Form: Textbooks
Document Type: Book
All Authors / Contributors: R K Bisht; H S Dhami
ISBN: 9780199452798 0199452792
OCLC Number: 932594511
Notes: Related resources available online.
Description: xxiv, 600 pages : illustrations ; 23 cm.
Contents: Machine generated contents note: 1.1. Discrete Mathematics-A Brief Introduction --
1.2. Introduction to Propositional Logic --
1.3. Proposition --
1.4. Logical Operators --
1.4.1. Negation (-) --
1.4.2. Disjunction (OR/v) --
1.4.3. Exclusive OR --
1.4.4. Conjunction (and/^) --
1.4.5. Conditional (->) --
1.4.6. Biconditional (<->) --
1.4.7. NAND (up arrow) --
1.4.8. NOR (down arrow) --
1.4.9. Well-formed Formula --
1.4.10. Rules of Precedence --
1.5. Tautology --
1.6. Contradiction --
1.7. Logical Equivalence --
1.8. Tautological Implication --
1.9. Converse, Inverse, and Contrapositive --
1.10. Functionally Complete Set of Connectives --
1.11. Normal Forms --
1.11.1. Elementary Product --
1.11.2. Elementary Sum --
1.11.3. Disjunctive Normal Form --
1.11.4. Conjunctive Normal Form --
1.11.5. Principal Disjunctive Normal Form --
1.11.6. Principal Conjunctive Normal Form --
1.12. Argument --
1.12.1. Checking the Validity of an Argument by Constructing Truth Table --
1.12.2. Checking the Validity of an Argument Without Constructing Truth Table --
1.13. Predicates --
1.13.1. Quantifiers --
1.13.2. Free and Bound Variables --
1.13.3. Negation of Quantifiers --
1.13.4. Removing Quantifiers from Predicates --
1.14. Nested Quantifiers --
1.14.1. Effect of Order of Quantifiers --
1.15. Inference Theory of Predicate Calculus --
1.15.1. Universal Specification --
1.15.2. Existential Specification --
1.15.3. Universal Generalization --
1.15.4. Existential Generalization --
1.15.5. Substitution --
1.15.6. First-order and Second-order Logic --
1.16. Methods of Proof --
1.16.1. Trivial Proof --
1.16.2. Vacuous Proof --
1.16.3. Direct Proof --
1.16.4. Proof by Contradiction --
1.16.5. Proof by Contraposition --
1.16.6. Proof by Cases --
1.16.7. Exhaustive Proof --
1.16.8. Proof by Mathematical Induction --
1.16.9. Proof by Minimal Counter Example --
1.17. Satisfiability and Consistency --
1.18. Mechanization of Reasoning --
1.18.1. Russell's Paradox --
2.1. Introduction --
2.2. Sets --
2.2.1. Roster Notation --
2.2.2. Set-builder Notation --
2.2.3. Cardinality of Sets --
2.3. Some Standard Sets --
2.3.1. Empty Set --
2.3.2. Singleton Set --
2.3.3. Finite and Infinite Sets --
2.3.4. Countable and Uncountable Sets --
2.3.5. Universal Set --
2.4. Subset and Proper Subset --
2.5. Equality of Sets --
2.6. Power Set --
2.7. Venn Diagrams --
2.8. Operations on Sets --
2.8.1. Union --
2.8.2. Intersection --
2.8.3. Difference of Two Sets --
2.8.4. Symmetric Difference of Two Sets --
2.8.5. Complement of a Set --
2.8.6. Generalized Union and Intersection --
2.9. Some Other Classes of Sets --
2.9.1. Disjoint Sets --
2.9.2. Partition --
2.9.3. Ordered Set --
2.9.4. Cartesian Product of Sets --
2.10. Algebra of Sets --
2.11. Multisets --
2.12. Fuzzy Sets --
2.12.1. Operations on Fuzzy Sets --
2.12.2. alpha- Cut and Strong alpha- Cut --
2.12.3. Support, Core, and Height of Fuzzy Sets --
3.1. Introduction --
3.2. Relation --
3.2.1. Domain and Range --
3.2.2. Inverse of Relation --
3.3. Combining Relations --
3.3.1. Composition of Relations --
3.4. Different Types of Relations --
3.4.1. Reflexive Relation --
3.4.2. Symmetric Relation --
3.4.3. Transitive Relation --
3.4.4. Compatible Relation --
3.4.5. Equivalence Relation --
3.4.6. Irreflexive Relation --
3.4.7. Asymmetric Relation --
3.4.8. Anti-symmetric Relation --
3.4.9. Partial Order Relation --
3.5. Pictorial or Graphical Representation of Relations --
3.6. Matrix Representation of Relations --
3.7. Closure of Relations --
3.7.1. Reflexive Closure --
3.7.2. Symmetric Closure --
3.7.3. Transitive Closure --
3.8. Warshall's Algorithm --
3.9. n-Ary Relations --
4.1. Introduction --
4.2. Definition of Function --
4.3. Relations Vs Functions --
4.4. Types of Functions --
4.4.1. One-One Function --
4.4.2. Many-One Function --
4.4.3. Onto Function --
4.4.4. Identity Function --
4.4.5. Constant Function --
4.4.6. Invertible Function --
4.5. Composition of Functions --
4.6. Sum and Product of Functions --
4.7. Functions Used in Computer Science --
4.7.1. Floor Function --
4.7.2. Ceiling Function --
4.7.3. Remainder Function/Modular Arithmetic --
4.7.4. Characteristic Function --
4.7.5. Hash Function --
4.8. Collision Resolution --
4.8.1. Open Addressing --
4.8.2. Chaining --
4.9. Investigation of Functions --
5.1. Introduction --
5.2. Basic Properties of Z --
5.3. Well-Ordering Principle --
5.4. Elementary Divisibility Properties --
5.5. Greatest Common Divisor --
5.6. Least Common Multiple --
5.7. Linear Diophantine Equation --
5.8. Fundamental Theorem of Arithmetic --
5.8.1. Primes and Composites --
5.8.2. Relatively Prime Integers --
5.9. Congruence Relation --
5.10. Residue Classes --
5.11. Linear Congruence --
6.1. Introduction --
6.2. Basic Counting Principle --
6.2.1. Sum Rule --
6.2.2. Product Rule --
6.2.3. Inclusion-Exclusion Principle --
6.3. Permutations and Combinations --
6.3.1. Permutation --
6.3.2. Combination --
6.4. Generalized Permutation and Combination --
6.4.1. Permutation with Repetition --
6.4.2. Permutations with Identical Objects --
6.4.3. Combination with Repetition --
6.5. Binomial Coefficients --
6.6. Partition --
6.7. Pigeonhole Principle --
6.7.1. Generalized Pigeonhole Principle --
6.8. Arrangements with Forbidden Positions --
6.8.1. Rook Polynomial --
6.8.2. Derangement --
7.1. Introduction --
7.2. Random Experiment --
7.3. Sample Space --
7.4. Event --
7.4.1. Equally Likely Events --
7.4.2. Mutually Exclusive Events --
7.4.3. Exhaustive Events --
7.4.4. Independent Events --
7.4.5. Dependent Events --
7.4.6. Complementary Event --
7.5. Measurement of Probability --
7.5.1. Classical or Priori Approach of Probability --
7.5.2. Relative Frequency Approach of Probability --
7.6. Axioms of Probability --
7.7. Conditional Probability --
7.8. Bayes' Theorem --
7.9. Discrete Probability Distributions --
7.9.1. Expectation of Random Variable --
7.9.2. Variance and Standard Deviation of Random Variables --
7.9.3. Binomial Distribution --
7.9.4. Poisson Distribution --
7.9.5. Negative Binomial Distribution --
7.9.6. Geometric Distribution --
8.1. Introduction --
8.2. Manipulation of Numeric Functions --
8.2.1. Sum and Product of Two Numeric Functions --
8.2.2. Multiplication with Scalar --
8.2.3. Modulus of Numeric Function --
8.2.4. Siar and S-iar of Numeric Function --
8.2.5. Forward and Backward Differences of Numeric Functions --
8.2.6. Accumulated Sum --
8.2.7. Convolution of Two Numeric Functions --
8.3. Generating Functions --
8.3.1. Properties of Generating Functions --
8.3.2. Solution of Combinatorial Problems Using Generating Functions --
9.1. Introduction --
9.2. Recursive Definition --
9.2.1. Recursively Defined Functions --
9.2.2. Recursively Defined Sets --
9.3. Recurrence Relation --
9.4. Solution of Recurrence Relations --
9.4.1. Iterative Method --
9.4.2. Recursive Method --
9.4.3. Generating Function --
9.5. Structural Induction --
9.6. Order and Degree of Recurrence Relations --
9.7. Linear Recurrence Relation with Constant Coefficients --
9.7.1. Linear Homogeneous Recurrence Relation with Constant Coefficients --
9.7.2. Linear Non-homogeneous Recurrence Relation with Constant Coefficients --
10.1. Introduction --
10.2. Binary Operations --
10.2.1. Semi-Group --
10.2.2. Monoid --
10.2.3. Group --
10.3. Addition and Multiplication Modulo m --
10.4. Subgroup --
10.4.1. Cosets --
10.5. Permutations and Symmetric Group --
10.5.1. Cyclic Permutation --
10.5.2. Stabilizer of an Element --
10.5.3. Orbit of an Element --
10.5.4. Invariant Elements under Permutation --
10.6. Cyclic Group --
10.7. Normal Subgroup --
10.8. Quotient Group --
10.9. Dihedral Group --
10.10. Homomorphism and Isomorphism --
10.10.1. Kernel of Homomorphism --
10.11. Ring --
10.11.1. Commutative Ring --
10.11.2. Ring with Unity --
10.11.3. Zero Divisor of a Ring --
10.11.4. Subrings --
10.11.5. Ring Homomorphism --
10.12. Integral Domain --
10.13. Division Ring or Skew Field --
10.14. Field --
10.15. Polynomial Ring --
10.16. Boolean Algebra --
10.16.1. Duality --
10.16.2. Boolean Functions --
10.16.3. Simplification of Boolean Functions --
10.16.4. Canonical Form --
10.16.5. Standard Form --
10.16.6. Other Logic Operations --
10.16.7. Karnaugh Map --
10.16.8. Quine-McCluskey Method --
10.16.9. Free Boolean Algebra --
11.1. Introduction --
11.2. Partially Ordered Set --
11.3. Diagrammatic Representation of Poset (Hasse Diagram) --
11.4. Elements in Posets --
11.4.1. Least and Greatest Elements --
11.4.2. Minimal and Maximal Elements --
11.4.3. Lower and Upper Bounds --
11.4.4. Greatest Lower Bound and Least Upper Bounds --
11.5. Linearly Ordered Set --
11.6. Well-Ordered Set --
11.7. Product Order --
11.8. Lexicographic Order --
11.9. Topological Sorting and Consistent Enumeration --
11.10. Isomorphism --
11.11. Lattices --
11.12. Properties of Lattices --
11.12.1. Principle of Duality --
11.12.2. Sublattice --
11.13. Some Special Lattices --
11.13.1. Modular Lattice --
11.13.2. Distributive Lattice --
11.13.3. Bounded Lattice --
11.13.4. Complemented Lattice --
11.13.5. Complete Lattice --
11.14. Product of Lattices --
11.15. Lattice Homomorphism Note continued: 11.16. Boolean Algebra and Lattices --
11.17. Stone's Representation Theorem --
12.1. Introduction --
12.2. Alphabet and Words --
12.3. Language --
12.4. Operations on Languages --
12.5. Finite Automata --
12.5.1. Deterministic Finite State Automata --
12.5.2. Non-Deterministic Finite Automata --
12.5.3. Conversion From Non-Deterministic Finite Automata to Deterministic Finite Automata --
12.5.4. Minimization of Finite Automata --
12.6. Finite Automata with Outputs --
12.6.1. Mealy Machine --
12.6.2. Moore Machine --
12.6.3. Equivalence of Mealy and Moore Machines --
12.6.4. Conversion From Mealy to Moore Machine --
12.6.5. Conversion From Moore to Mealy Machine --
12.7. Regular Expression --
12.8. Regular Expression and Finite Automata --
12.9. Generalized Transition Graph --
12.10. Grammar of Formal Languages --
12.10.1. Phrase Structure Grammar --
12.10.2. Chomsky Hierarchy --
12.11. Other Machines --
13.1. Introduction --
13.2. Graph and its Related Definitions --
13.3. Different Types of Graphs --
13.3.1. Simple Graph --
13.3.2. Multigraph, Trivial Graph, and Null Graph --
13.3.3. Complete Graph --
13.3.4. Regular Graph --
13.3.5. Bipartite Graph --
13.3.6. Weighted Graph --
13.4. Subgraphs --
13.5. Operations on Graphs --
13.5.1. Union of Two Graphs --
13.5.2. Intersection of Two Graphs --
13.5.3. Ring Sum of Two Graphs --
13.5.4. Decomposition of a Graph --
13.5.5. Deletion of a Vertex --
13.5.6. Deletion of an Edge --
13.5.7. Complement of a Graph --
13.6. Walk, Path, and Circuit --
13.6.1. Walk --
13.6.2. Path --
13.6.3. Circuit --
13.7. Connected Graph, Disconnected Graph, and Components --
13.8. Homomorphism and Isomorphism of Graphs --
13.9. Homeomorphic Graphs --
13.10. Euler and Hamiltonian Graphs --
13.10.1. Euler Line and Euler Graph --
13.10.2. Hamiltonian Path and Hamiltonian Circuit --
13.10.3. Travelling Salesman Problem --
13.11. Planar Graph --
13.11.1. Kuratowski's Two Graphs --
13.11.2. Region and its Degree --
13.11.3. Euler's Formula --
13.12. Tree --
13.12.1. Rooted Tree --
13.12.2. Binary Tree --
13.12.3. Height of Binary Tree --
13.12.4. Spanning Tree --
13.12.5. Branch and Chord --
13.12.6. Rank and Nullity --
13.12.7. Fundamental Circuits --
13.12.8. Finding All Spanning Trees of a Graph --
13.12.9. Spanning Trees in a Weighted Graph --
13.12.10. Kruskal's Algorithm --
13.12.11. Prim's Algorithm --
13.12.12. Dijkstra Algorithm --
13.12.13. Binary Search Tree --
13.13. Cut Set and Cut Vertex --
13.14. Colouring of Graphs --
13.14.1. Chromatic Number --
13.14.2. Chromatic Partitioning --
13.14.3. Independence Set and Maximal Independence Set --
13.14.4. Maximum Independence Set and Independence Number --
13.14.5. Clique and Maximal Clique --
13.14.6. Maximum Clique and Clique Number --
13.14.7. Perfect Graph --
13.14.8. Chromatic Polynomial --
13.14.9. Applications of Graph Colouring --
13.15. Matching --
13.15.1. Maximal Matching, Maximum Matching, and Matching Number --
13.15.2. Perfect Matching --
13.16. Matrix Representation of Graphs --
13.16.1. Incidence Matrix --
13.16.2. Circuit Matrix --
13.16.3. Cut Set Matrix --
13.16.4. Path Matrix --
13.16.5. Adjacency Matrix --
13.17. Traversal of Graphs --
13.17.1. Breadth-First Search --
13.17.2. Depth-First Search --
13.18. Traversing Binary Trees --
13.18.1. Pre-Order Traversal --
13.18.2. In-Order Traversal --
13.18.3. Post-Order Traversal --
13.19. Digraph or Directed Graph --
13.20. Network Flow --
13.20.1. Cut in a Transport Network --
13.20.2. Flow Augmenting Path --
13.21. Enumeration of Graphs --
14.1. Introduction --
14.2. Asymptotic Behaviour of Numeric Functions --
14.2.1. Big-Oh (O) Notation --
14.2.2. Omega Notation --
14.2.3. Theta Notation --
14.3. Analysis of Algorithms --
14.3.1. Space Complexity --
14.3.2. Time Complexity --
14.4. Analysis of Sorting Algorithms --
14.4.1. Insertion Sort --
14.4.2. Bubble Sort --
14.4.3. Selection Sort --
14.5. Divide-and-Conquer Approach --
14.5.1. Merge Sort --
14.5.2. Quick Sort --
14.6. Analysis of Searching Algorithms --
14.6.1. Linear Search --
14.6.2. Binary Search --
14.7. Tractable and Intractable Problems --
14.8. Logic Gates --
14.8.1. Switching Circuits and Logic Gates --
14.8.2. NAND and NOR Implementations --
14.9. Combinational Circuits --
14.9.1. Half Adder --
14.9.2. Full Adder --
14.9.3. Half Subtractor --
14.9.4. Full Subtractor --
14.10. Information and Coding Theory --
14.10.1. Discrete Information Sources --
14.10.2. Entropy --
14.10.3. Mutual Information --
14.10.4. Coding Theory --
14.10.5. Hamming Distance --
14.10.6. Error-Detecting and Error-Correcting Codes --
14.10.7. Group Codes --
14.10.8. Generator Matrices --
14.10.9. Parity Check Matrices --
14.10.10. Coset Decoding --
14.10.11. Prefix Codes --
14.10.12. Cyclic Code.
Series Title: Oxford higher education.
Responsibility: R.K. Bisht (Associate Professor, Department of Computer Science and Applications, Amrapali Group of Institutes, Haldwani (Uttarakhand)), H.S. Dhami (Vice-Chancellor, Kumaun University, Nainital).

Abstract:

Discrete Mathematics is a textbook designed for the students of computer science engineering, information technology, and computer applications to help them develop the foundation of theoretical  Read more...

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