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PARADOX

PARADOX. Hui Hiu Yi Kary Chan Lut Yan Loretta Choi Wan Ting Vivian. CHINESE SAYINGS FOR RELATIONSHIPS …. 好馬不吃回頭草 好草不怕回頭吃. BECAUSE…. P( 好馬 ) ¬ P( 壞馬 ) Q( 好草 ) ¬ Q( 壞草 ) P → ¬ (Q ˅ ¬ Q) (P ˅ ¬ P) → Q. a ‘visual’ paradox: Illusion. Falsidical paradox.

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PARADOX

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  1. PARADOX Hui Hiu Yi Kary Chan Lut Yan Loretta Choi Wan Ting Vivian

  2. CHINESE SAYINGSFOR RELATIONSHIPS … • 好馬不吃回頭草 • 好草不怕回頭吃

  3. BECAUSE…. • P(好馬) ¬ P(壞馬) Q(好草) ¬Q(壞草) • P→¬ (Q˅¬Q) • (P ˅ ¬ P) →Q

  4. a ‘visual’ paradox: Illusion

  5. Falsidical paradox • A proof that sounds right, but actually it is wrong! • Due to: Invalid mathematical proof logical demonstrations of absurdities

  6. Example 1: 1=0 (?!) • Let x=0 • x(x-1)=0 • x-1=0 • x=1 • 1=0 What went wrong?

  7. EXAMPLE 2: THE MISSING SQUARE (?!)

  8. EXAMPLE 3A: ALL MATH162 STUDENTS ARE OF THE SAME GENDER! • We need 5 boys!

  9. Example 3b: All angry birds are the same in colour! • Supposed we have 5 angry birds of unknown color. How to prove that they all have the same colour? 1 2 3 4 5

  10. If we can proofthat, 4 of them are the same color…. E.g. 1 2 3 4 are the same color ... and another 4 of them are also red (including the previous excluded one,5) 4 4 4 E.g. 2 3 4 5are same color 1 1 1 2 2 2 Then 5 of them must be the same! 3 3 3 5 5 5

  11. Hence, how can we prove that 4 of them are the same too? • We can use the same logic!

  12. If we can proofthat, 3 of them are the same…. E.g. 1 2 3 are the same ... and another 3 of them are also the same (including the previous excluded one,4) 4 4 4 E.g. 2 3 4 are red 1 1 1 2 2 2 Then 4 of them must be the same! 3 3 3

  13. But we know, not all angry birds are the same in color • What went wrong? • Hint: Can we continue the logic proof from 5 angry birds to 1 angry bird? Why? Why not?

  14. BARBER PARADOX(BERTRAND RUSSELL, 1901) • Barber(n.) = hair stylist • Once upon a time... There is a town... - no communication with the rest of the world - only 1 barber - 2 kinds of town villagers: - Type A: people who shave themselves - Type B: people who do not shave themselves - The barber has a rule: He shaves Type B people only.

  15. QUESTION:WILL HE SHAVE HIMSELF? • Yes. He will! • No. He won't! • Which type of people does he belong to?  What’s Wrong ?

  16. ANTINOMY (二律背反) • p -> p' and p' -> p • p if and only if not p • Logical Paradox • More examples:  • (1) Liar Paradox • "This sentence is false."  Can you state one more example for that paradox? • (2) Grelling-Nelson Paradox • "Is the word 'heterological' heterological?"  • heterological(adj.) = not describing itself • (3) Russell's Paradox:  • next slide....

  17. RUSSELL'S PARADOX • Discovered by Bertrand Russell at 1901 • Found contradiction on Naive Set Theory • What is Naive Set Theory?  • Hypothesis: If x is a member of A, x ∈ A. • e.g. Apple is a member of Fruit, Apple ∈ Fruit • Contradiction: 

  18. Birthday Paradox • How many people in a room, that the probability of at least two of them have the same birthday, is more than 50%? • Assumption: • No one born on Feb 29 • No Twins • Birthdays are distributed evenly

  19. Calculation Time • Let O(n) be the probability of everyone in the room having different birthday, where n is the number of people in the room • O(n) = 1 x (1 – 1/365) x (1 – 2/365) x … x [1 - (n-1)/365] • O(n) = 365! / 365n (365 – n)! • Let P(n) be the probability of at least two people sharing birthday • P(n) = 1 – O(n) = 1 - 365! / 365n (365 – n)!

  20. Calculation Time (Cont.) • P(n) = 1 - 365! / 365n (365 – n)! • P(n) ≥ 0.5 → n = 23

  21. Why it is a paradox? • No logical contradiction • Mathematical truth contradicts Native Intuition • Veridical Paradox Application?

  22. Birthday Attack • Well-known cryptographic attack • Crack a hash function What is hash function?

  23. Hash Function • Use mathematical operation to convert a large, varied size of data to small datum • Generate unique hash sum for each input • For security reason (e.g. password) • MD5 (Message-Digest algorithm 5)

  24. MD5 • One of widely used hash function • Return 32-digit hexadecimal number for each input • Usage: Electronic Signature, Password • Unique Fingerprint However …

  25. Security Problems • Infinite input But finite hash sum • Different inputs may result same hash sum (Hash Collision !!!) • Use forged electronic signature • Hack other people accounts How to get hash collision?

  26. Try every possible inputs • Possible hash sum: 1632 = 3.4 X 1038 • 94 characters on a normal keyboard • Assume the password length is 20 • Possible passwords: 9420 = 2.9 X 1039

  27. Birthday Attack • P(n) = 1 - 365! / 365n (365 – n)! • A(n) = 1 - k! / kn (k– n)! where kis maximum number of password tried • A(n) = 1 – e^(-n2/2k) • n = √2k ln(1-A(n)) • Let the A(n) = 0.99 • n = √-2(3.4 X 1038) ln(1-0.99) = 5.6 x 1019 • 5.6 x 1019 = 1.93 x 10-20 original size

  28. 3 Type of Paradox • Veridical Paradox: contradict with our intuition but is perfectly logical • Falsidical paradox: seems true but actually is false due to a fallacy in the demonstration. • Antinomy: be self-contradictive

  29. HOMEWORK • 1. Please state two sentences, so that Prof. Li will give you an A in MATH162. (Hints: The second sentence can be: Will you give me an A in MATH162?) • 2. Consider the following proof of 2 = 1 • Let a = b • a2 = ab • a2 – b2 = ab – ab2 • (a-b)(a+b) = b(a-b) • a + b = b • b + b = b • 2b = b • 2 = 1 • Which type of paradox is this? Which part is causing the proof wrong?

  30. EXTRA CREDIT • Can you find another example of paradox and crack it?

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