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 Tabel kebenaran Tautologi dan kontradiksi 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: Maka, 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

Operator dan Tabel Kebenaran Konjungsi p  q diucapkan “p dan q”  p  q Bernilai salah jika salah satu pernyataan salah p q p  q 1

Operator dan Tabel Kebenaran Disjungsi Inklusif p  q diucapkan “p atau q”  p  q Bernilai benar jika salah satu pernyataan benar p q p  q 1

Operator dan Tabel Kebenaran Disjungsi Eksklusif p  q Bernilai benar jika hanya salah satu dari p dan q bernilai benar p q p  q 1

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

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 q p  q 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)  (p  q)]  q  [p  (p  q)]  q  [ F  (p  q)]  q  (p  q)  q  (p  q)  q  (p  q)  q  p  (q  q )  p  T  T substitution for  distributive 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 T F p q  p q  p T F

Propositional Logic - proof of 1 famous  I could say “prove a law of distributivity.” Distributivity: p  (q  r)  (p  q)  (p  r) p q r q  r p  (q  r) p  q p  r (p  q)  (p  r) T F All truth assignments for p, q, and r.

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 “If I am hungry, then it is noon.” Inverses: p  q and p  q “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???

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 )

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”

Contoh 2 Let p denote “The automated reply can be sent” 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

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.

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:

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.

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

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

Axiom, postulates, hypotheses and previously proven theorems. 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

What rule of inference can we use to justify it? 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 Inference Rule: Modus Ponens Tautology: (p  (p  q))  q

Proofs - Modus Tollens I am not a great skater. If I am Erik, then I am a great skater.  I am not Erik! Inference Rule: Modus Tollens q p  q  p Tautology: (q  (p  q))  p

Proofs - Addition I am not a great skater.  I am not a great skater or I am tall. p  p  q Tautology: p  (p  q) Inference Rule: Addition

Proofs - Simplification I am not a great skater and you are sleepy.  you are sleepy. p  q  p Tautology: (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: Hypothetical Syllogism Tautology: ((p  q)  (q  r))  (p  r) p  q q  r  p  r

Proofs - Exercise Amy is a computer science major.  Amy is a math major or a computer science major. Addition 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. It does not rain.  It is not cloudy Denying the hypothesis. 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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