PARADOX

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PARADOX

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

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CHINESE SAYINGS FOR RELATIONSHIPS

• ???????
• ???????

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BECAUSE.

• P(??) P(??) Q(??) Q(??)
• P ? (Q ? Q)
• (P ? P) ? Q

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a visual paradox Illusion
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Falsidical paradox

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

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Example 1 10 (?!)

• Let x0
• x(x-1)0
• x-10
• x1
• 10

What went wrong?
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EXAMPLE 2 THE MISSING SQUARE (?!)
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(No Transcript)
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EXAMPLE 3A ALL MATH162 STUDENTS ARE OF THE SAME
GENDER!

• We need 5 boys!

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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?

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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!
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Hence, how can we prove that 4 of them are the
same too?

• We can use the same logic!

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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!
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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?

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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.

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QUESTIONWILL HE SHAVE HIMSELF?

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

Whats Wrong ?
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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....

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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 

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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

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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)!

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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
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Why it is a paradox?

• No logical contradiction
• Mathematical truth contradicts Native Intuition
• Veridical Paradox

Application?
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Birthday Attack

• Well-known cryptographic attack
• Crack a hash function

What is hash function?
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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)

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MD5

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

However
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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?
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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

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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

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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

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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?

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EXTRA CREDIT

• Can you find another example of paradox and crack
  it?