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

2. CHINESE SAYINGS FOR 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 : I LLUSION

5. F ALSIDICAL PARADOX A proof that sounds right, but actually it is wrong! Due to: Invalid mathematical proof logical demonstrations of absurdities

6. E XAMPLE 1: 1=0 (?!) Let x=0 x(x-1)=0 x-1=0 x=1 1=0

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

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

10. Supposed we have 5 angry birds of unknown color. How to prove that they all have the same colour? E XAMPLE 3 B : A LL ANGRY BIRDS ARE THE SAME IN COLOUR ! 1 2 3 4 5

11. E.g. 1 2 3 4 are the same color 1234 5... and another 4 of them are also red (including the previous excluded one, 5) E.g. 2 3 4 5 are same color 1234 5 Then 5 of them must be the same! 1234 5 If we can proof that, 4 of them are the same color….

12. H ENCE, HOW CAN WE PROVE THAT 4 OF THEM ARE THE SAME TOO ? We can use the same logic!

13. E.g. 1 2 3 are the same 1234... and another 3 of them are also the same (including the previous excluded one, 4) E.g. 2 3 4 are red 1234 Then 4 of them must be the same! 1234 If we can proof that, 3 of them are the same….

14. B UT 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?

15. 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.

16. QUESTION: WILL HE SHAVE HIMSELF? Yes. He will! No. He won't! Which type of people does he belong to?

17. 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....

18. 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:

19. B IRTHDAY P ARADOX How many people in a room, that the probability of at least two of them have the same birthday, is more than 50%? Assumption: 1. No one born on Feb 29 2. No Twins 3. Birthdays are distributed evenly

20. C ALCULATION T IME 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! / 365 n (365 – n)! Let P(n) be the probability of at least two people sharing birthday P(n) = 1 – O(n) = 1 - 365! / 365 n (365 – n)!

21. C ALCULATION T IME (C ONT.) P(n) = 1 - 365! / 365 n (365 – n)! P(n) ≥ 0.5 → n = 23 nP(n) 1011.7% 2041.1% 2350.7% 3070.6% 5097.0% 5599.0%

22. W HY IT IS A PARADOX ? No logical contradiction Mathematical truth contradicts Native Intuition Veridical Paradox

23. B IRTHDAY A TTACK Well-known cryptographic attack Crack a hash function

24. H ASH F UNCTION 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)

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

26. S ECURITY P ROBLEMS Infinite input But finite hash sum Different inputs may result same hash sum (Hash Collision !!!) Use forged electronic signature Hack other people accounts

27. T RY EVERY POSSIBLE INPUTS Possible hash sum: 16 32 = 3.4 X 10 38 94 characters on a normal keyboard Assume the password length is 20 Possible passwords: 94 20 = 2.9 X 10 39

28. B IRTHDAY A TTACK P(n) = 1 - 365! / 365 n (365 – n)! A(n) = 1 - k! / k n (k – n)! where k is maximum number of password tried A(n) = 1 – e^(-n 2 /2k) n = √2k ln(1-A(n)) Let the A(n) = 0.99 n = √-2(3.4 X 10 38 ) ln(1-0.99) = 5.6 x 10 19 5.6 x 10 19 = 1.93 x 10 -20 original size

29. 3 T YPE OF P ARADOX 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

30. 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 a 2 = ab a 2 – b 2 = ab – ab 2 (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?

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