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PARADOX

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PARADOX Hui Hiu Yi Kary Chan Lut Yan Loretta Choi Wan Ting Vivian – PowerPoint PPT presentation

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Title: PARADOX


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

2
CHINESE SAYINGS FOR RELATIONSHIPS
  • ???????
  • ???????

3
BECAUSE.
  • P(??) P(??) Q(??) Q(??)
  • 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 10 (?!)
  • Let x0
  • x(x-1)0
  • x-10
  • x1
  • 10

What went wrong?
7
EXAMPLE 2 THE MISSING SQUARE (?!)
8
(No Transcript)
9
EXAMPLE 3A ALL MATH162 STUDENTS ARE OF THE SAME
GENDER!
  • We need 5 boys!

10
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?

11
If we can proof that, 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)
E.g. 2 3 4 5 are same color
Then 5 of them must be the same!
12
Hence, how can we prove that 4 of them are the
same too?
  • We can use the same logic!

13
If we can proof that, 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)
E.g. 2 3 4 are red
Then 4 of them must be the same!
14
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?

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
QUESTIONWILL HE SHAVE HIMSELF?
  • Yes. He will!
  • No. He won't!
  • Which type of people does he belong to? 

Whats Wrong ?
17
ANTINOMY (????)
  • p -gt p' and p' -gt 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
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

20
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)!

21
Calculation Time (Cont.)
  • P(n) 1 - 365! / 365n (365 n)!
  • P(n) 0.5 ? n 23

n P(n)
10 11.7
20 41.1
23 50.7
30 70.6
50 97.0
55 99.0
22
Why it is a paradox?
  • No logical contradiction
  • Mathematical truth contradicts Native Intuition
  • Veridical Paradox

Application?
23
Birthday Attack
  • Well-known cryptographic attack
  • Crack a hash function

What is hash function?
24
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)

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

However
26
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?
27
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

28
Birthday Attack
  • P(n) 1 - 365! / 365n (365 n)!
  • A(n) 1 - k! / kn (k n)! where k is maximum
    number of password tried
  • A(n) 1 e(-n2/2k)
  • n v2k ln(1-A(n))
  • Let the A(n) 0.99
  • n v-2(3.4 X 1038) ln(1-0.99) 5.6 x 1019
  • 5.6 x 1019 1.93 x 10-20 original size

29
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

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
  • a2 ab
  • a2 b2 ab ab2
  • (a-b)(ab) 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?
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