PARADOXHui 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

A proof that sounds right, but actually it is wrong!

Due to:Invalid mathematical prooflogical demonstrations of absurdities

EXAMPLE 1: 1=0 (?!)

Let x=0 x(x-1)=0 x-1=0 x=1 1=0

EXAMPLE 2: THE MISSING SQUARE (?!)

EXAMPLE 3A: ALL MATH162 STUDENTS ARE OF THE SAME GENDER!

We need 5 boys!

Supposed we have 5 angry birds of unknown color. How to prove that they all have the same colour?

EXAMPLE 3B: ALL ANGRY BIRDS ARE THE SAME IN COLOUR!

1 2

3 4 5

E.g. 1 2 3 4 are the same color 1 2 3 4 5

... and another 4 of them are also red (including the previous excluded one, 5)

E.g. 2 3 4 5 are same color

1 2 3 4 5

Then 5 of them must be the same!

1 2 3 4 5

If we can proof that, 4 of them are the same color….

HENCE, HOW CAN WE PROVE THAT 4 OF THEM ARE THE SAME TOO?

We can use the same logic!

E.g. 1 2 3 are the same 1 2 3 4

... and another 3 of them are also the same (including the previous excluded one, 4) E.g. 2 3 4 are red 1 2 3 4

Then 4 of them must be the same!1 2 3 4

If we can proof that, 3 of them are the same….

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?

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.

QUESTION:WILL HE SHAVE HIMSELF?

Yes. He will! No. He won't!

Which type of people does he belong to? 

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....

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: 

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:1. No one born on Feb 292. No Twins3. Birthdays are distributed evenly

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)!

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%

WHY IT IS A PARADOX?

No logical contradiction

Mathematical truth contradicts Native Intuition

Veridical Paradox

BIRTHDAY ATTACK

Well-known cryptographic attack

Crack a hash function

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)

MD5 One of widely used hash function Return 32-digit hexadecimal number for each

input Usage: Electronic Signature, Password Unique Fingerprint

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

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

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 = √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

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

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?

EXTRA CREDIT

Can you find another example of paradox and crack it?