METHOD OF PROOF 030513122 Discrete Mathematics Asst Prof

METHOD OF PROOF 030513122 - Discrete Mathematics Asst. Prof. Dr. Choopan Rattanapoka

ทำไมตองพสจน (1) “Mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning. ” -John Locke Mathematical proofs are, in a sense, the only true knowledge we have They provide us with a guarantee as well as an explanation (and hopefully some insight)

Modus Ponens (MP) Modus Ponens ( -elimination) P Q P Q T T F F F T T F F T

Addition (Add) Addition ( -introduction) P P Q หรอ Q P Q P Q P Q T T T T F T T F F F

Simplification (Simp) Simplification ( -elimination) P Q P หรอ P Q Q P Q P Q T T T T F F F F F

Conjunction (Conj) Conjunction ( -introduction) P Q P Q T T F F F T F F

Rules of Inference: Resolution For resolution, we have the following tautology ((p q) ( p r)) (q r) Essentially, If we have two true disjunctions that have mutually exclusive propositions Then we can conclude that the disjunction of the two non-mutually exclusive propositions is true

ทบทวน Rule of References อกนด Affirming the antecedent: Modus ponens (p q)) q Denying the consequent: Modus Tollens ( q (p q)) p Affirming the conclusion: Fallacy (q (p q)) p Denying the hypothesis: Fallacy ( p (p q)) q

ตวอยาง G(x): “x is in the garage” E(x): “x has an engine problem” S(x): “x has been sold” Universe of discourse คอ ดงนนจะไดสมมตฐานทวา Quantifier จงแสดงวาเมอรวา “A car in the garage has an engine problem” และ “Every car in the garage has been sold” สามารถสรปไดวา “A car has been sold has an engine problem” กำหนด : การพสจนแบบม รถทงหมด : x (G(x) E(x)) x (G(x) S(x)) ขอสรปทตองการคอ x (S(x) E(x))

Proofs with Quantifiers: Example (2) 1. 2. 3. 4. 5. 6. 7. 8. 9. x (G(x) E(x)) (G(c) E(c)) G(c) x (G(x) S(x)) G(c) S(c) E(c) S(c) E(c) x (S(x) E(x)) 1 st premise (1) Existential Instantiation (2) Simp 2 nd premise (4) Universal Instantiation (3), (5) MP (2) Simp (6), (7) Conj (8) Existential generalization

ทำแบบฝกหดดวยกนกอนพกครง (1) จากขอความตอไปน มการใช rule of inference อะไรบาง Alice is a mathematics major. Therefore, Alice is either a mathematics major or a computer science major. Jerry is a mathematics major and a computer science major. Therefore, Jerry is a mathematics major. If it is rainy, then the pool will be closed. It is rainy. Therefore, the pool is closed. If it snows today, the university will close. The university is not closed today. Therefore, it did not snow today. If I go swimming, then I will stay in the sun too long. If I stay in the sun too long, then I will sunburn. Therefore, if I go swimming, then I will sunburn.

วธการ Proofs Trivial proofs Vacuous proofs Direct proofs Proof by Contrapositive (indirect proof) Proof by Contradiction (indirect proof, aka refutation) Proof by Cases (sometimes using WLOG) Proofs of equivalence Existence Proofs (Constructive & Nonconstructive)