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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