matdis 1.1-1.2
TRANSCRIPT
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Matematika Diskrit3 SKS
Buku Teks :
Discrete Mathematics and Its Applications,Kenneth H Rosen, McGraw-Hill, 6th edition
Penilaian :
tugas : 20%
tes 1 : 25%
tes 2 : 25%
uas : 30%
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discrete mathematics
1. The foundations : Logic and Proofs
2. Basic Structures: sets, functions, sequences, sums
3. The Fundamentals: algorithms, the integers, matrices
4.
Induction and Recursion5. Counting
6. Discrete Probability
7. Advanced Counting Techniques
8. Relations9. Graphs
10. Trees
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discrete mathematics
11. Boolean Algebra
12. Modeling Computation
13. Appendices
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Chapter 1The Foundations: Logic and Proofs
1.1. Propositional Logic
1.2. Propositional Equivalences
1.3. Predicates and Quantifiers
1.4. Nested Quantifiers1.5. Rules of Inference
1.6. Introduction to Proofs
1.7. Proof Methods and Strategy
End-of-Chapter Material
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Propositional Logic
Chapter 1.1.
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Proposition A propositional variable is denoted by the letters p, q, r,
It is either true or false, but not both
Its true value is called true (1) or false (0)
Propositional variables are denoted by the letters p, q, etc.
Examples : today is Tuesday
1 + 1 = 2
2 + 2 = 3
Not a proposition: what time is it ?
you may be seated
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Compound Propositions
compound statements
Let p, q, r be simple propositions
A compound proposition is obtained by
connecting p, q, r using logical operators
(or connectives)
Example: we are studying and it is rainingSurabaya is a city or Malang is an ocean
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connectives
NOT (negation) Symbol
AND (conjunction) Symbol
Inclusive OR (disjunction) Symbol v
EXclusive OR (XOR) Symbol
Conditional statement Symbol
(implication)
Biconditional Symbol
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Level of Precedence
NEGATION (NOT)
CONJUNCTION (AND)
DISJUNCTION (OR, XOR)
CONDITIONAL
BICONDITIONAL
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examples
compound propositions examples:
(p q) r
p (q r)
( p) ( q)
(p q) ( r)
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Truth Table
Negation
example: p = today is Tuesday
p = today is not Tuesday
(today is Monday)
p p
0 1
1 0
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Truth Table
conjunctionp q p q
0 0 0
0 1 01 0 0
1 1 1
example: p = today is Tuesday
q = it is raining
p q = today is Tuesday and it is raining
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Truth table
disjunction (inclusive or)
example: p = John is a studentq = Mia is a lawyer
p v q = John is a student or Mia is a lawyer
p q p v q
0 0 0
0 1 11 0 1
1 1 1
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Truth table
p q p q
0 0 0
0 1 11 0 1
1 1 0
example: p = John is a student
q = Mia is a lawyerp v q = either John is a student or Mia is a lawyer
exclusive or
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Truth Table (p r) q
p q r (p r) q
0 0 0 (0 1) 0 = 0
0 0 1 (0 0) 0 = 0
0 1 0 (0 1) 1 = 1
0 1 1 (0 0 1 = 1
1 0 0 (1 1) 0 = 1
1 0 1 (1 0) 0 = 0
1 1 0 (1 1) 1 = 11 1 1 (1 0) 1 = 1
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Truth Table p r q
p q r p (r q)
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 01 1 1
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Example 18 p. 13a logic puzzle
by Smullyan
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Truth Table
implication
p q p q
0 0 1
0 1 1
1 0 0
1 1 1
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Implication
Notation : p q
Examples :
1. if 2 + 2 = 4 then x := x + 1
2. if m > 0 then y := 2 * y
3. if it is raining then we will not go
Let s denote 2 + 2 = 4 and a denote x := x + 1
The symbolic notation for example 1 : s a
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Hypothesis & Conclusion
In the implication p q
p is called the antecedent, hypothesis, premise
q is called the consequence,conclusion
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Ways to express p q
jika p maka q if p then q
jika p, q if p, q
q jika p q if p
p hanya jika q p only if q
p mengimplikasikan q p implies q
see page 6
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Necessary & Sufficient conditions
p qis necessaryfor p
is a sufficientcondition for q
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Conversion & Inversion
The conversion of p q is q p The inversion of p q is p q
p q is not equivalent to q p
p
q is not equivalent to
p
q
p q p q q p p q
0 0 1 1 1
0 1 1 0 0
1 0 0 1 1
1 1 1 1 1
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contrapositive
The contrapositive of p q is q p. p q and q p are equivalent
p q p q q p
0 0 1 10 1 1 1
1 0 0 0
1 1 1 1
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Biconditional
p if and only if q
p q
p q p q (p q) (q p)
0 0 1 1
0 1 0 0
1 0 0 0
1 1 1 1
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Propositional Equivalence
Chapter 1.2.
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Tautology
A proposition that is always true
example: p p v q
p q p p v q
0 0 1
0 1 1
1 0 1
1 1 1
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Contradiction
a proposition that is always false
example : p ( p )
p p ( p)
0 0
1 0
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Logical Equivalence
Notation p q ( p and q are compound propositions )
Example : p q is logically equivalent to p q
p q p q p q
0 0 1 1
0 1 1 1
1 0 0 0
1 1 1 1
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Logical Equivalence
See pages 24, 25
Table 6
Table 7
Table 8
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De Morgans Law
(p q) ( p) ( q)
(p q) ( p) ( q)
p q p q p q (p q) ( p) ( q)
0 0 1 1 0 1 1
0 1 1 0 1 0 0
1 0 0 1 1 0 0
1 1 0 0 1 0 0
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HomeworkChapter 1.1. no. 35 - 38
Chapter 1.2. no. 7, 9, 16, 17