Note for Discrete Mathematics - DMS by bala vijay

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MA8351 Discrete Mathematics www.padeepz.net A Course Material on de ep z .n et MA8351 - Discrete Mathematics By Ms. M.KAVITHA ASSISTANT PROFESSOR DEPARTMENT OF SCINENCE AND HUMANITIES .p a SASURIE COLLEGE OF ENGINEERING w w w VIJAYAMANGALAM – 638 056 www.padeepz.net

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MA8351 Discrete Mathematics www.padeepz.net CONTENT S. No. 3.1 3.2 3.3 3.4 4.1 4.2 4.3 4.4 ne t 1 1 7 12 13 16 18 23 32 33 49 53 61 67 71 88 91 94 143 152 156 163 182 w w w 5.1 5.2 5.3 UNIT – 1 LOGIC AND PROOFS INTRODUCTION LOGICAL CONNECTIVES PROPOSITIONAL EQUIVALENCE PREDICATES & QUANTIFIERS RULES OF INFERENCE INTRODUCTION TO PROOFS METHODS AND STRATEGY UNIT II COMBINATORICS MATHEMATICAL INDUCTION STRONG INDUCTION & WELL ORDERING RECURRENCE RELATION SOLVING LINEAR RECURRENCE RELATIONS GENERATING FUNCTION THE PRINCIPLE OF INCLUSION & EXCLUSION UNIT III GRAPHS GRAPH & GRAPH MODELS GRAPH TERMINOLOGY SPECIAL TYPES OF GRAPHS EULER &HAMILTONIAN GRAPH UNIT IV -ALGEBRAIC SYSTEMS ALGEBRAIC SYSTEMS SEMIGROUPS-MONOIDS- HOMOMORPHISM GROUPS –SUBGROUPS& HOMOMORPHISM COSET & LAGRANGES THEOREM- ALGEBRAIC SYSTEMS WITH TWO BINARY OPERATIONS UNIT V – LATTICES & BOOLEAN ALGEBRA PARTIAL ORDERING- POSETS- LATTICES PROPERTIES OF LATTICESBOOLEAN ALGEBRA pz . 2.1 2.2 2.3 2.4 2.5 2.6 Page No. .p ad ee 1.1 1.2 1.3 1.4 1.5 1.6 TOPIC www.padeepz.net

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MA8351 Discrete Mathematics www.padeepz.net Unit I 1.1 INTRODUCTION ad ee pz .n et LOGIC AND PROOFS PROPOSITION (OR) STATEMENT: Proposition is a declarative statement that is either true or false but not both. The truth value of proposition is true or false. Truth table It displays the relationship between the truth values of proposition. Negation of a proposition If P is a proposition, then its negation is denoted by ¬P or ~p and is defined by the following truth table. P F T w w .p T F ¬P EXAMPLE P - Ram is intelligent ¬P -Ram is not intelligent proposition is a declarative sentence which is either true or false but not both. COMPOUND PROPOSITION w It is a proposition consisting of two or more simple proposition using logical operators. 1.2 LOGICAL CONNECTIVES (1) DISJUNCTION (OR) The disjunction of two proposition P and Q is the proposition P˅Q defined by the following truth table. P Q P˅Q [read as P or Q ] and is www.padeepz.net

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MA8351 Discrete Mathematics T T F F T F T F www.padeepz.net T T T F (1) CONJUNCTION (AND) ad ee pz .n et If P and Q are two propositions , then the conjunction of P and Q is denoted by P˄Q ( read as P and Q ) and is defined by following truth table. P Q P˄Q T T T T F F F T F F F F CONDITIONAL AND BI- CONDITIONAL PROPOSITION (1) Conditional proposition If p and q are propositions, then the implication “If p then q “ denoted by p→q , called the conditional statement of p and q , is defined by following truth table. p T T F F NOTE q T F T F p→q T F T T w w .p p→q is false when p is true and q is false. Otherwise it is true. The different situations where the conditional statements applied are listed below. (1) If p then q (2) p only if q (3) q whenever p (4) q is necessary for p (5) q follows from p (6) q when p (7) p is sufficient for q (8) p implies q w Converse, contrapositive and Inverse statement If p→q is a conditional statement, then (1) q→p is called converse of p→q (2) ¬q→¬p is called contrapositive of p→q (3) ¬p→¬q is called inverse of p→q EXAMPLE p : Ram is a computer science student q : Ram study DBMS www.padeepz.net