Foundations of Discrete MathematicsThis Book Is Meant To Be More Than Just A Text In Discrete Mathematics. It Is A Forerunner Of Another Book Applied Discrete Structures By The Same Author. The Ultimate Goal Of The Two Books Are To Make A Strong Case For The Inclusion Of Discrete Mathematics In The Undergraduate Curricula Of Mathematics By Creating A Sequence Of Courses In Discrete Mathematics Parallel To The Traditional Sequence Of Calculus-Based Courses.The Present Book Covers The Foundations Of Discrete Mathematics In Seven Chapters. It Lays A Heavy Emphasis On Motivation And Attempts Clarity Without Sacrificing Rigour. A List Of Typical Problems Is Given In The First Chapter. These Problems Are Used Throughout The Book To Motivate Various Concepts. A Review Of Logic Is Included To Gear The Reader Into A Proper Frame Of Mind. The Basic Counting Techniques Are Covered In Chapters 2 And 7. Those In Chapter 2 Are Elementary. But They Are Intentionally Covered In A Formal Manner So As To Acquaint The Reader With The Traditional Definition-Theorem-Proof Pattern Of Mathematics. Chapters 3 Introduces Abstraction And Shows How The Focal Point Of Todays Mathematics Is Not Numbers But Sets Carrying Suitable Structures. Chapter 4 Deals With Boolean Algebras And Their Applications. Chapters 5 And 6 Deal With More Traditional Topics In Algebra, Viz., Groups, Rings, Fields, Vector Spaces And Matrices.The Presentation Is Elementary And Presupposes No Mathematical Maturity On The Part Of The Reader. Instead, Comments Are Inserted Liberally To Increase His Maturity. Each Chapter Has Four Sections. Each Section Is Followed By Exercises (Of Various Degrees Of Difficulty) And By Notes And Guide To Literature. Answers To The Exercises Are Provided At The End Of The Book. |
Contents
Introduction and Preliminaries | 1 |
Elementary Counting Techniques | 53 |
Sets with Additional Structures | 129 |
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a₁ abelian group algebraic structures apply b₁ bijection binary operation Boolean algebra Boolean function boxes called cardinality Chapter circuit coefficient commutative complex numbers concept continuous mathematics coset cyclic defined definition denoted discrete mathematics distinct equals equation equivalence relation euclidean example exists expressed field F follows function f given gives group G Hence Hint homomorphism identity element induction infinite integral domain inverse isomorphic last exercise Let G linear transformation linearly independent matrix modulo multiplication n₁ non-zero normal subgroup Note pair partial order partitions permutation polynomial positive integer power series prime problem proof properties Proposition Prove quotient group r₁ real numbers recurrence relation result ring root sequences of length Similarly simply solution solving statement subgroup of G subring subset Suppose switches symmetric Theorem tion unique V₁ values vector space x₁ Y₁