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)

P → ¬ (Q ˅ ¬Q)

(P ˅ ¬ P) → Q
Q(好草)
¬Q(壞草)
A ‘VISUAL’ PARADOX: ILLUSION
FALSIDICAL PARADOX


A proof that sounds right, but actually it is wrong!
Due to:
Invalid mathematical proof
logical 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!
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
3
2
4
5
If we can proof that, 4 of them are the same
color….
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
HENCE, HOW CAN WE PROVE THAT 4 OF
THEM ARE THE SAME TOO?

We can use the same logic!
If we can proof that, 3 of them are the
same….
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
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.
2.
3.
No one born on Feb 29
No Twins
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 - (n1)/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
10
20
23
30
50
55
P(n)
11.7%
41.1%
50.7%
70.6%
97.0%
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?