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MA8351
Discrete Mathematics
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A Course Material on
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MA8351 - Discrete Mathematics
By
Ms. M.KAVITHA
ASSISTANT PROFESSOR
DEPARTMENT OF SCINENCE AND HUMANITIES
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SASURIE COLLEGE OF ENGINEERING
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VIJAYAMANGALAM – 638 056
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MA8351
Discrete Mathematics
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CONTENT
S. No.
3.1
3.2
3.3
3.4
4.1
4.2
4.3
4.4
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12
13
16
18
23
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33
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61
67
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143
152
156
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182
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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
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2.1
2.2
2.3
2.4
2.5
2.6
Page No.
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1.1
1.2
1.3
1.4
1.5
1.6
TOPIC
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MA8351
Discrete Mathematics
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Unit I
1.1 INTRODUCTION
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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
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.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
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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
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MA8351
Discrete Mathematics
T
T
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F
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T
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F
(1) CONJUNCTION (AND)
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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
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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
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F
F
NOTE
q
T
F
T
F
p→q
T
F
T
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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
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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
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