Download CS311H: Discrete Mathematics Cardinality of Infinite Sets and

Announcements
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and
Introduction to Number Theory
Instructor: Işıl Dillig
Instructor: Işıl Dillig,
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
1/33
Homework 4 due now
I
Midterm next lecture; covers everything up to this lecture
I
Recall: Midterm closed-book, closed-notes, but can bring
3-pages of hand-prepared notes
Instructor: Işıl Dillig,
Cardinality of Infinite Sets
I
I
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
2/33
Example
Sets with infinite cardinality are classified into two classes:
Prove: The set of odd positive integers is countably infinite.
1. Countably infinite sets (e.g., natural numbers)
I
Need to find a function f from Z+ to the set of odd positive
integers, and prove that f is bijective
I
Consider f (n) = 2n − 1 from Z+ to odd positive integers
I
We need to show f is bijective (i.e., one-to-one and onto)
I
Let’s first prove injectivity, then surjectivity
2. Uncountably infinite sets (e.g., real numbers)
I
A set A is called countably infinite if there is a bijection
between A and the set of positive integers.
I
A set A is called countable if it is either finite or countably
infinite
I
Otherwise, the set is called uncountable or uncountably infinite
Instructor: Işıl Dillig,
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
3/33
Instructor: Işıl Dillig,
Example, cont.
Instructor: Işıl Dillig,
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
4/33
Another Way to Prove Countable-ness
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
5/33
I
One way to show a set A is countably infinite is to give
bijection between Z+ and A
I
Another way is by showing members of A can be written as a
sequence (a1 , a2 , a3 , . . .)
I
Since such a sequence is a bijective function from Z+ to A,
writing A as a sequence a1 , a2 , a3 , . . . establishes one-to-one
correspondence
Instructor: Işıl Dillig,
1
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
6/33
Another Example
Rational Numbers are Countable
Prove that the set of all integers is countable
I
We can list all integers in a sequence, alternating positive and
negative integers:
an = 0, 1, −1, 2, −2, 3, −3, . . .
I
Instructor: Işıl Dillig,
Observe that this sequence defines the bijective function:
n/2
if n even
f (n) =
−(n − 1)/2 if n odd
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
7/33
Set of rationals is also countably infinite!
I
We’ll prove that the set of positive rational numbers is
countable by showing how to enumerate them in a sequence
I
Recall: Every positive rational number can be written as the
quotient p/q of two positive integers p, q
Instructor: Işıl Dillig,
Rationals in a Table
I
I
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
8/33
Enumerating the Rationals
Now imagine placing rationals in a table such that:
1. Rationals with p = 1 go in first row, p = 2 in second row, etc.
2. Rationals with q = 1 in 1st column, q = 2 in 2nd column, . . .
Instructor: Işıl Dillig,
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
First list those with p + q = 2, then p + q = 3, . . .
I
Traverse table diagonally from left-to-right, in the order shown
by arrows
Instructor: Işıl Dillig,
9/33
Enumerating the Rationals, cont.
I
I
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
10/33
Uncountability of Real Numbers
This allows us to list all rationals in a sequence:
I
Prime example of uncountably infinite sets is real numbers
I
Proven by George Cantor using the famous diagonalization
argument
I
For contradiction, assume the set of reals was countable
I
Since any subset of a countable set is also countable, this
would imply the set of reals between 0 and 1 is also countable
1 2 1 1 2 3 4 3
, , , , , , , ,...
1 1 2 3 2 1 1 2
I
Instructor: Işıl Dillig,
Hence, set of rationals is countable
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
Instructor: Işıl Dillig,
11/33
2
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
12/33
Diagonalization Argument
I
Diagonalization Argument, concluded
Now, we’ll create a new real number R and show that it is not
equal to any of the Ri ’s in this sequence:
I
Let R = 0.a1 a2 a3 . . . such that:
4 dii 6= 4
ai =
5 dii = 4
I
Clearly, this new number R differs from each number Ri in
the table in at least one digit (its i ’th digit)
Instructor: Işıl Dillig,
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
Number theory is the branch of mathematics that deals with
integers and their properties
I
Number theory has a number of applications in computer
science, esp. in modern cryptography
Next few lectures: Basic concepts in number theory and its
application in crypto in cryptography
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
Hence, the set of real between 0 and 1 is not countable
I
Since the superset of any uncountable set is also uncountable,
set of reals is uncountably infinite
I
If a divides b, we write a|b; otherwise, a 6 | b
I
Example: 2|6, 2 6 | 9
I
If a|b, a is called a factor of b
I
b is called a multiple of a
14/33
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
16/33
Properties of Divisibility
I
I
Recall: All positive integers divisible by d are of the form dk
I
I
We want to find how many numbers dk there are such that
0 < dk ≤ n.
I
I
In other words, we want to know how many integers k there
are such that 0 < k ≤ nd
Instructor: Işıl Dillig,
Cardinality of Infinite Sets and Introduction to Number Theory
Given two integers a and b where a 6= 0, we say a divides b if
there is an integer c such that b = ac
Question: If n and d are positive integers, how many positive
integers not exceeding n are divisible by d ?
I
CS311H: Discrete Mathematics
I
Instructor: Işıl Dillig,
15/33
Example
I
I
Divisibility
I
Instructor: Işıl Dillig,
Since R is not in the table, this is not a complete enumeration
of all reals between 0 and 1
Instructor: Işıl Dillig,
13/33
Introduction to Number Theory
I
I
How many integers are there between 1 and
CS311H: Discrete Mathematics
Theorem 1: If a|b and b|c, then a|c
I
I
n
d?
Cardinality of Infinite Sets and Introduction to Number Theory
17/33
Instructor: Işıl Dillig,
3
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
18/33
Divisibility Properties, cont.
I
Theorem 2: If a|b and a|c, then a|(mb + nc) for any int m, n
I
Proof:
I
Corollary 1: If a|b and a|c, then a|(b + c) for any int c
I
Corollary 2: If a|b, then a|mb for any int m
Instructor: Işıl Dillig,
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
The Division Theorem
Division theorem: Let a be an integer, and d a positive
integer. Then, there are unique integers q, r with 0 ≤ r < d
such that a = dq + r
I
Here, d is called divisor, and a is called dividend
I
q is the quotient, and r is the remainder.
I
We use the r = a mod d notation to express the remainder
I
The notation q = a div d expresses the quotient
I
What is 101 mod 11?
I
What is 101 div 11?
Instructor: Işıl Dillig,
19/33
Congruence Modulo
I
I
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
Congruence Modulo Theorem
In number theory, we often care if two integers a, b have same
remainder when divided by m.
I
Theorem: a ≡ b (mod m) iff a mod m = b mod m
I
Part 1, ⇒: Suppose a ≡ b (mod m).
I
If so, a and b are congruent modulo m, a ≡ b (mod m).
I
Then, by definition of ≡, m|(a − b)
I
More technically, if a and b are integers and m a positive
integer, a ≡ b (mod m) iff m|(a − b)
I
By definition of |, there exists k such that a − b = mk , i.e.,
a = b + mk
I
Example: 7 and 13 are congruent modulo 3.
I
By division thm, b = mp + r for some 0 ≤ r < m
I
Example: Find a number congruent to 7 modulo 4.
I
Then, a = mp + r + mk = m(p + k ) + r
I
Thus, a mod m = r = b mod m
Instructor: Işıl Dillig,
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
21/33
Instructor: Işıl Dillig,
Congruence Modulo Theorem Proof, cont.
I
Theorem: a ≡ b (mod m) iff a mod m = b mod m
I
Part 2, ⇐: Suppose a mod m = b mod m
I
Then, there exists some p1 , p2 , r such that a = p1 · m + r and
b = p2 · m + r where 0 ≤ r < m
I
22/33
Prove that if a ≡ b (mod m) and c ≡ d (mod m), then:
I
I
I
I
Thus, m|(a − b)
I
I
By definition of ≡, a ≡ b (mod m)
I
Cardinality of Infinite Sets and Introduction to Number Theory
Cardinality of Infinite Sets and Introduction to Number Theory
a + c ≡ b + d (mod m)
Then, a − b = p1 · m + r − p2 · m − r = m · (p1 − p2 )
CS311H: Discrete Mathematics
CS311H: Discrete Mathematics
Example
I
Instructor: Işıl Dillig,
20/33
23/33
Instructor: Işıl Dillig,
4
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
24/33
Applications of Congruence in Cryptography
I
Congruences have many applications in cryptography, e.g.,
shift ciphers
I
Shift cipher with key k encrypts message by shifting each
letter by k letters in alphabet (if past Z , then wrap around)
I
What is encryption of ”KILL HIM” with shift cipher of key 3?
I
Shift ciphers also called Ceasar ciphers because Julius Ceasar
encrypted secret messages to his generals this way
Mathematical Encoding of Shift Ciphers
I
First, let’s number letters A-Z with 0 − 25
I
Represent message with sequence of numbers
I
Example: The sequence ”25 0 2” represents ”ZAC”
I
To encrypt, apply encryption function f defined as:
f (x ) = (x + k ) mod 26
I
Because f is bijective, its inverse yields decryption function:
g(x ) = (x − k ) mod 26
Instructor: Işıl Dillig,
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
25/33
Instructor: Işıl Dillig,
Ciphers and Congruence Modulo
I
Shift cipher is a very primitive and insecure cipher because
very easy to infer what k is
I
But contains some useful ideas:
I
I
Encoding words as sequence of numbers
I
Use of modulo operator
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
27/33
Fundamental Thm: Every positive integer greater than 1 is
either prime or can be written uniquely as a product of primes.
I
This unique product of prime numbers for x is called the
prime factorization of x
I
Examples:
Instructor: Işıl Dillig,
I
A positive integer p that is greater than 1 and divisible only
by 1 and itself is called a prime number.
I
First few primes: 2, 3, 5, 7, 11, . . .
I
A positive integer that is greater than 1 and that is not prime
is called a composite number
I
Example: 4, 6, 8, 9, . . .
Instructor: Işıl Dillig,
Fundamental Theorem of Arithmetic
I
Cardinality of Infinite Sets and Introduction to Number Theory
26/33
Prime Numbers
Modern encryption schemes much more sophisticated, but
also share these principles (coming lectures)
Instructor: Işıl Dillig,
CS311H: Discrete Mathematics
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
28/33
Determining Prime-ness
I
In many applications, such as crypto, important to determine
if a number is prime – following thm is useful for this:
I
Theorem: If n is composite, then it has a prime divisor less
√
than or equal to n
I
I
12 =
I
I
21 =
I
I
99 =
I
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
29/33
Instructor: Işıl Dillig,
5
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
30/33
Consequence of This Theorem
Theorem: If n is composite, then it has a prime divisor ≤
Infinitely Many Primes
√
n
I
Thus, to determine if n is prime, only need to check if it is
√
divisible by primes ≤ n
I
Example: Show that 101 is prime
I
I
Instructor: Işıl Dillig,
I
Theorem: There are infinitely many prime numbers.
I
Proof: (by contradiction) Suppose there are finitely many
primes: p1 , p2 , . . . , pn
I
Now consider the number Q = p1 p2 . . . pn + 1. Q is either
prime or composite
I
Case 1: Q is prime. We get a contradiction, because we
assumed only prime numbers are p1 , . . . , pn
I
Case 2: Q is composite. In this case, Q can be written as
product of primes.
I
But Q is not divisible by any of p1 , p2 , . . . , pn
I
Hence, by Fundamental Thm, not composite ⇒ ⊥
√
Since 101 < 11, only need to check if it is divisible by
2, 3, 5, 7.
Since it is not divisible by any of these, we know it is prime.
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
31/33
Instructor: Işıl Dillig,
6
CS311H: Discrete Mathematics
Cardinality of Infinite Sets and Introduction to Number Theory
32/33