Pembuktian Matematika Matematika Diskrit Semester

Metode Pembuktian
Matematika Diskrit
Logika…

Logika merupakan studi penalaran
◦ Penalaran adalah cara berpikir dengan
mengembangkan sesuatu berdasarkan akal
budi
◦ Bukan karena perasaan atau pengalaman

Filsuf Yunani => Aristoteles => 2300
tahun yang lalu
Review
Semua pengendara sepeda motor
memakai helm
 Setiap orang yang memakai helm adalah
mahasiswa
 Jadi, semua pengendara sepeda motor
adalah mahasiswa

Logika…

Review
◦ Proposisi
 Konjungsi, Disjungsi, Ingkaran (Negasi), Implikasi,
Biimplikasi
◦ Tabel kebenaran
◦ Tautologi dan kontradiksi
Proposisi

Proposisi : kalimat yang bernilai B atau S
◦ Ekspresi yang mengandung VARIABEL berupa
pernyataan dan OPERATOR sebagai konektor
VARIABEL :Variabel Proposisional
 OPERATOR : Operator Logika

Contoh








6 adalah bilangan genap
Ibukota Jawa barat adalah Semarang
2+2=4
Kehidupan hanya ada di planet bumi
Serahkan uangmu sekarang!
x+3=8
Siapa yang menjadi presiden Indonesia?
x>3
Operator





NEGASI
: TIDAK (NOT) => ~
KONJUNGSI : DAN (AND) => ∧
DISJUNGSI
: ATAU (OR) => ∨
KONDISIONAL
: Jika … Maka => →
BIKONDISIONAL : Jika dan hanya jika => ↔
Kombinasi proposisi

Contoh:
◦ p : hari ini hujan
◦ q : hari ini dingin

Maka,
◦ q ∨ ~p : hari ini dingin atau hari ini tidak hujan
◦ = : hari ini dingin atau tidak hujan
Latihan
p: pemuda itu tinggi
 q: pemuda itu tampan
 Buat pernyataan berikut dalam bentuk
ekspresi logika:

◦ Pemuda itu tinggi tapi tidak tampan
◦ Tidak benar bahwa pemuda itu pendek atau
tidak tampan
◦ Pemuda itu tinggi atau pendek dan tampan
◦ Tidak benar bahwa pemuda itu pendek
maupun tampan
Operator dan Tabel Kebenaran

Negasi/Ingkaran/Peniadaan/Not
 ~p diucapkan “tidaklah p”
 Jika P benar maka ~P salah
p
~p
1
0
0
1
Operator dan Tabel Kebenaran

Konjungsi
◦ p  q diucapkan “p dan q”  p  q
◦ Bernilai salah jika salah satu pernyataan salah
p
1
1
0
0
q
1
0
1
0
pq
1
0
0
0
Operator dan Tabel Kebenaran

Disjungsi Inklusif
◦ p  q diucapkan “p atau q”  p  q
◦ Bernilai benar jika salah satu pernyataan benar
p
1
1
0
0
q pq
1
1
0
1
1
1
0
0
Operator dan Tabel Kebenaran

Disjungsi Eksklusif
◦pq
◦ Bernilai benar jika hanya salah satu dari p dan
q bernilai benar
p
1
1
0
0
q pq
1
0
0
1
1
1
0
0
Operator dan Tabel Kebenaran

Kondisional/Bersyarat/Implikasi
◦
◦
◦
◦
p  q  p  q diucapkan :
“jika p maka q” atau “p hanya jika q” atau
“p cukup untuk q” atau “q perlu untuk p”
Bernilai benar kecuali jika P benar dan Q salah
p q pq
1 1
1
1 0
0
0 1
1
0 0
1
Operator dan Tabel Kebenaran

Bikondisional/Biimplikasi
◦ p  q diucapkan “p jika dan hanya jika q”
◦ Bernilai benar jika pernyataan/nilai
kebenarannya sama, jika berbeda maka
menjadi salah
p
1
1
0
0
q pq
1
1
0
0
1
0
0
1
Penggunaan operator di google
Hukum-hukum logika proposisi
Mathematical Logic

Tautology
◦ A statement formula A is said to be a tautology if
the truth value of A is T for any assignment of the
truth values T and F to the statement variables
occurring in A

Contradiction
◦ A statement formula A is said to be a
contradiction if the truth value of A is F for any
assignment of the truth values T and F to the
statement variables occurring in A
Propositional Logic – an unfamous 
if NOT (blue AND NOT red) OR red then…
(p  q)  q  p  q
(p  q)  q

(p  q)  q
DeMorgan’s

(p  q)  q
Double negation

p  (q  q)
Associativity

p  q
Idempotent
Propositional Logic - one last proof

Show that [p  (p  q)]  q is a tautology.

We use  to show that [p  (p  q)]  q  T.
[p  (p  q)]  q
 [p  (p  q)]  q
substitution for 
 [(p  p)  (p  q)]  q
distributive
 [ F  (p  q)]  q
 (p  q)  q
 (p  q)  q
 (p  q)  q
 p  (q  q )
 p  T
T
uniqueness
identity
substitution for 
DeMorgan’s
associative
excluded middle
domination
Propositional Logic - logical equivalence
Challenge: Try to find a proposition that is equivalent to p  q,
but that uses only the connectives , , and .
p
q
pq
p
q
p
q  p
T
T
F
F
T
F
T
F
T
F
T
T
T
T
F
F
T
F
T
F
F
F
T
T
T
F
T
T
Propositional Logic - proof of 1 famous 
Distributivity:
p  (q  r)  (p  q)  (p  r)
I could say
“prove a law of
distributivity.”
p
q
r
qr
p  (q  r)
pq
pr
(p  q)  (p  r)
T
T
T
T
T
T
T
T
T
T
F
F
T
T
T
T
T
F
T
F
T
T
T
T
T
F
F
F
T
T
T
T
F
T
T
T
T
T
T
T
F T F
All truth
F F T
assignment
F F
s forFp, q,
and r.
F
F
T
F
F
F
F
F
T
F
F
F
F
F
F
Propositional Logic - special definitions
Hint: In one
instance, the pair
of propositions is
equivalent.
Contrapositives: p  q and q  p
 Ex. “If it is noon, then I am hungry.”
“If I am not hungry, then it is not noon.”
Converses: p  q and q  p
 Ex. “If it is noon, then I am hungry.”
“If I am hungry, then it is noon.”
Inverses: p  q and p  q
 Ex. “If it is noon, then I am hungry.”
“If it is not noon, then I am not hungry.”
p  q  q  p
Menterjemahkan bahasa ke dalam bentuk
ekspresi logika matematika
English is often ambiguous and translating
sentences into compound propositions
removes the ambiguity.
Using logical expressions, we can analyze them
and determine their truth values. And we can
use rules of inferences to reason about them.
Bagaimana dengan bahasa Indonesia???
24
Contoh 1
“ You can access the internet from campus
only if you are a computer science major
or you are not a freshman.”
p : “You can access the internet from
campus”
q : “You are a computer science major”
r : “You are freshmen”
p  ( q v ~r )
25
Spesifikasi Sistem
Translating sentences in natural language into
logical expressions is an essential part of
specifying both hardware and software
systems.
 Consistency of system specification.
 Spesifikasi:

◦ “The automated reply cannot be sent when
the file system is full”
26
Contoh 2
Let p denote “The automated reply can be
sent”
2. Let q denote “The file system is full”
The logical expression for the sentence
“The automated reply cannot be sent when
the file system is full” is
1.
27
Contoh 3
Determine whether these system specifications
are consistent:
 The diagnostic message is stored in the
buffer or it is retransmitted.
 The diagnostic message is not stored in the
buffer.
 If the diagnostic message is stored in the
buffer, then it is retransmitted.
28
Contoh 3 …
Let p denote “The diagnostic message is
stored in the buffer”
 Let q denote “The diagnostic message is
retransmitted”

The three specifications are:
29
Contoh 4
If we add one more requirement
 “The diagnostic message is not
retransmitted”
The new specifications now are

Sistem tidak konsisten karena tidak ada nilai BENAR pada
statement tersebut.
Dapat dibuktikan pada tabel kebenaran.
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Metode Pembuktian
Aturan Inferensia
 Fallacy : Bentuk pembuktian yang salah

◦ Fallacy of Affirming the Conclusion
◦ Fallacy of Denying the Hypothesis
◦ Circular Reasoning

Metode pembuktian teorema (implikasi)
◦ Vacuous Proof dan Trivial Proof
(Bukti Kosong dan Trivial)
◦ Direct dan Indirect Proof
(Bukti Langsung dan Tidak Langsung)
Inference and Substitution
32
33
Aturan Inferensia
Contoh

Budi adalah mahasiswa Untar dan tinggal di
Jakarta. Budi tinggal di Jakarta.
◦ Aturan inferensianya : Simplification

Jika ani pergi berenang maka ani akan
berjemur matahari. Jika ani pergi berjemur
matahari maka ani akan terbakar matahari.
Maka jika ani pergi berenang maka ani akan
terbakar matahari.
◦ Aturan inferensianya : Hypothetical Syllogism
Proofs - how do you know?
A theorem is a statement that can be
shown to be true.
A proof is the means of doing so.
Axiom, postulates,
hypotheses and
previously proven
theorems.
Rules of inference
Proof
Proofs - how do you know?
The following statements are true:
If I am Mila, then I am a great swimmer.
I am Mila.
What do we know to be true?
I am a great swimmer!
What rule of
inference can we use
to justify it?
Proofs - Modus Ponens
I am Mila.
If I am Mila, then I am a great swimmer.
 I am a great swimmer!
p
pq
q
Tautology:
Inference
Rule:
(p  (p  q))  q
Modus Ponens
Proofs - Modus Tollens
I am not a great skater.
If I am Erik, then I am a great skater.
 I am not Erik!
q
pq
 p
Tautology:
(q  (p  q))  p
Inference
Rule:
Modus Tollens
Proofs - Addition
I am not a great skater.
 I am not a great skater or I am tall.
p
pq
Tautology:
Inference
Rule:
p  (p  q)
Addition
Proofs - Simplification
I am not a great skater and you are sleepy.
 you are sleepy.
Tautology:
pq
p
(p  q)  p
Inference Rule:
Simplification
Proofs - Disjunctive Syllogism
I am a great eater or I am a great skater.
I am not a great skater.
 I am a great eater!
pq
q
p
Tautology:
((p  q)  q)  p
Inference
Rule:
Disjunctive
Syllogism
Proofs - Hypothetical Syllogism
If you are an athlete, you are always hungry.
If you are always hungry, you have a snickers in
your backpack.
 If you are an athlete, you have a snickers in
your backpack.
Inference Rule:
pq
Tautology:
qr
((p  q)  (q  r))  (p  r)
pr
Hypothetical
Syllogism
Proofs - Exercise
Amy is a computer science major.
Addition
 Amy is a math major or a computer science
major.
If Ernie is a math major then Ernie is geeky.
Ernie is not geeky!
 Ernie is not a math major.
Modus Tollens
Proofs - Fallacies
Rules of inference, appropriately
applied give valid arguments.
Mistakes in applying rules of inference
are called fallacies.
Proofs - valid arg or fallacy?
Affirming the
conclusion.
If I am Bonnie Blair, then I skate fast
I skate fast!
 I am Bonnie Blair
I’m Eric Heiden
((p  q)  q)  p
Not a tautology.
If you don’t give me $10, I bite your ear.
I bite your ear!
 You didn’t give me $10.
I’m just mean.
Proofs - valid arg or fallacy?
If it rains then it is cloudy.
Denying the
hypothesis.
It does not rain.
 It is not cloudy
February!
((p  q)  p)  q
Not a tautology.
If it is a car, then it has 4 wheels.
It is not a car.
 It doesn’t have 4 wheels.
ATV
Metode Pembuktian Teorema

Proof by Implication
◦ Vacuous Proof
◦ Trivial Proof
Vacuous Proof

is a truth that is devoid of content because it
asserts something about all members of a
class that is empty or because it says
"If A then B" when in fact A is inherently
false.
Direct Proof

In mathematics and logic, a direct proof is a
way of showing the truth or falsehood of a
given statement by a straightforward
combination of established facts, usually
existing lemmas and theorems, without
making any further assumptions.
Indirect Proof (Proof by Contradiction)

Indirect proof is a type of proof in which a
statement to be proved is assumed false and
if the assumption leads to an impossibility,
then the statement assumed false has been
proved to be true.
Referensi
Rosen
 Wikipedia
 iCoachMath.com

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