1 Introduction to Abstract Mathematics Chapter 3: Elementary Number Theory and Methods of Proofs...
-
Upload
miles-scot-neal -
Category
Documents
-
view
222 -
download
0
Transcript of 1 Introduction to Abstract Mathematics Chapter 3: Elementary Number Theory and Methods of Proofs...
1Introduction to Abstract Mathematics
Chapter 3: Elementary Number Theory and Methods of Proofs
Instructor: Hayk Melikya [email protected]
3.1-.3.4 Direct Methods and Counterexamples• Introduction• Rational Numbers• Divisibility• Division Algorithm
2Introduction to Abstract Mathematics
Basic Definitions
Definition: An integer n is an even number if there exists an integer k such that n = 2k.
Def: An integer n is an odd number if there exists an integer k such that n = 2k+1.
Def: An integer n is a prime number if and only if n>1
and if n=rs for some positive integers r and s then r=1 or s=1.
Symbolically: Let Even(n) := “an integer n is even”: E(n) = ( k Z)( n = 2k) .
Symbolically: Let O(n) := “an integer is odd”: Odd(n) = ( k Z)( n = 2k +1) .
3Introduction to Abstract Mathematics
Primes and CompositesDef: An integer n is a prime number if and only if n>1
and if n=rs for some positive integers r and s then r=1 or s=1.
Symbolically: Prime(n):= n is prime
positive integers r and s, if n = rs then r =1 or s =1
Def: A positive integer n is a composite if and only if n=rs for some
positive integers r and s then r ≠ 1 and s ≠ 1.
Symbolically: Cpmposite(n):= n is compoeite positive integers r and s,
such that n = rs and r ≠ 1 and s ≠ 1
The RSA Challenge (up to US$200,000)http://www.rsasecurity.com/rsalabs/challenges/factoring/numbers.html
Examples: Find the truth value of the following prpopositionsE(6), P(12), C(17)
4Introduction to Abstract Mathematics
Existential Statements
x P(x)Proofs:
– Constructive Construct an example of such a such that P(a) is true
– Non-constructive By contradiction
– Show that if such x does NOT exist than a contradiction can be derived
5Introduction to Abstract Mathematics
Example
Let G(n):= a b ((a+b=n) Prime(a) Prime(b))
Prove that (nN)G(n) Proof:
– n=210 – a=113– b=97
Piece of cake…
What about (nN) G(n) ( many Million $ baby)
6Introduction to Abstract Mathematics
Universal Statements
x P(x) x [Q(x) R(x)]
Proof techniques:– Exhaustion– By contradiction
Assume the statement is not true Arrive at a contradiction
– Direct Generalizing from an arbitrary particular member
7Introduction to Abstract Mathematics
├ (xU)P(x)
To prove a theorem of the form (xU)P(x) (same as ├ (xU)P(x))
which states “for all elements x in a given universe U, the
proposition P(x) is true” we select an arbitrary aU from the
universe, and then prove the assertion P(a).
Then by Universal generalization we conclude P(a)├ (xU)P(x)
For arguments of the form├ x [Q(x) R(x)]
8Introduction to Abstract Mathematics
Example 1
Exhaustion:– Any even number between 4 and 30 can be written as a
sum of two primes:– 4=2+2– 6=3+3– 8=3+5– …– 30=11+19
Works for finite domains only
What if I want to prove that for any integer n the product of n and n+1 is even?
Can I exhaust all integer values of n?
9Introduction to Abstract Mathematics
Example 2
Theorem: (nZ)( even(n*(n+1)) )
Proof:– Consider a particular but arbitrary chosen integer n – n is odd or even– Case 1: n is odd
Then n=2k+1, n+1=2k+2 n(n+1) = (2k+1)(2k+2) = 2(2k+1)(k+1) = 2p for some
integer p So n(n+1) is even
Case 2: n is evenThen n=2k, n+1=2k+1 n(n+1) = 2k(2k+1) = 2p for some integer pSo n(n+1) is even
10Introduction to Abstract Mathematics
Fallacy
Generalizing from a particular but NOT arbitrarily chosen example
I.e., using some additional properties of nExample:
– “all odd numbers are prime”– “Proof”:
Consider odd number 3 It is prime Thus for any odd n prime(n) holds
Such “proofs” can be given for correct statements as well!
11Introduction to Abstract Mathematics
Prevention
Try to stay away from specific instances (e.g., 3) Make sure that you are not using any additional properties of n
considered Challenge your proof
– Try to play the devil’s advocate and find holes in it…
• Using the same letter to mean different things• Jumping to a conclusion• Insufficient justification• Begging the question assuming the claim first
12Introduction to Abstract Mathematics
Rational Numbers
A real number is rational iff it can be representedas a ratio/quotient/fraction of integers a and b(b0)
rR [rQ a,bZ [r=a/b & b0]]
Notes:– a is numerator– b is denominator– Any rational number can be represented in infinitely many
ways– The fractional part of any rational number written in any
natural radix has a period in it
13Introduction to Abstract Mathematics
Rational or not? -12
– -12/1 3.1459
– 3+1459/10000 0.56895689568956895689…
– 5689/9999 1+1/2+1/4+1/8+…
– 2 0
– 0/1
14Introduction to Abstract Mathematics
Theorem 1
Any number with a periodic fractional part in a natural
radix representation is rational
Proof:– Constructive:
– x=0.n1…nmn1…nm…
– x=0.(n1…nm)
– x*10m-x=n1…nm
– x=n1…nm/(10m-1)
15Introduction to Abstract Mathematics
Theorem 2
Any geometric series:– S=q0+q1+q2+q3+…– where -1<q<1– evaluates to S=1/(1-q)
Proof– Proof idea – More formal proof– Definitions of limits and partial sums
16Introduction to Abstract Mathematics
Z Q
Every integer is a rational numberProof : set the denominator to 1
Book : page 127
The set of rational numbers is closed with respect to arithmetic operations +, -, *, /
Partial proofs : textbook pages 121-131Formal proof
Q
17Introduction to Abstract Mathematics
Irrational Numbers
So far all the examples were of rational numbers
How about some irrationals? – e– sqrt(2)
18Introduction to Abstract Mathematics
Simple Exercises
The sum of two even numbers is even.
The product of two odd numbers is odd.
direct proof.
19Introduction to Abstract Mathematics
a “divides” b or is b divisible by a (a|b ):
b = ak for some integer k
Also we say that
b is multiple of a
a is a factor of b
b is divisor for a
Divisibility
5|15 because 15 = 35
n|0 because 0 = n0
1|n because n = 1n
n|n because n = n1
A number p > 1 with no positive integer divisors other than 1 and itself
is called a prime. Every other number greater than 1 is called
composite. 2, 3, 5, 7, 11, and 13 are prime,
4, 6, 8, and 9 are composite.
20Introduction to Abstract Mathematics
1. If a | b, then a | bc for all c.
2. If a | b and b | c, then a | c.
3. If a | b and a | c, then a | sb + tc for all s and t.
4. For all c ≠ 0, a | b if and only if ca | cb.
Simple Divisibility Facts
Proof of (??)
direct proof.
21Introduction to Abstract Mathematics
Divisibility by a Prime
Theorem. Any integer n > 1 is divisible by a prime number.
22Introduction to Abstract Mathematics
Every integer, n>1, has a unique factorization into primes:
p0 ≤ p1 ≤ ··· ≤ pk
p0 p1 ··· pk = n
Fundamental Theorem of Arithmetic
Example:
61394323221 = 3·3·3·7·11·11·37·37·37·53
23Introduction to Abstract Mathematics
Claim: Every integer > 1 is a product of primes.
Prime Products
Proof: (by contradiction)
Suppose not. Then set of non-products is nonempty.
There is a smallest integer n > 1 that is not a product of
primes.
In particular, n is not prime.
So n = k·m for integers k, m where n > k,m >1.
Since k,m smaller than the least nonproduct, both are prime
products, eg.,
k = p1 p2 p94
m = q1 q2 q214
24Introduction to Abstract Mathematics
Prime Products
…So
n = k m = p1 p2 p94 q1 q2 q214
is a prime product, a contradiction.
The set of nonproducts > 1 must be empty. QED
Claim: Every integer > 1 is a product of primes.
(The proof of the fundamental theorem will be given later.)
25Introduction to Abstract Mathematics
For b > 0 and any a, there are unique numbers
q : quotient(a,b), r : remainder(a,b), such that
a = qb + r and 0 r < b.
The Quotient-Reminder Theorem
When b=2, this says that for any a,
there is a unique q such that a=2q or a=2q+1.
When b=3, this says that for any a,
there is a unique q such that a=3q or a=3q+1 or a=3q+2.
We also say q = a div b r = a mod b.
26Introduction to Abstract Mathematics
For b > 0 and any a, there are unique numbers
q : quotient(a,b), r : remainder(a,b), such that
a = qb + r and 0 r < b.
The Division Theorem
0 b 2b kb (k+1)b
Given any b, we can divide the integers into many blocks of b numbers.
For any a, there is a unique “position” for a in this line.
q = the block where a is in
a
r = the offset in this block
Clearly, given a and b, q and r are uniquely defined.
-b
27Introduction to Abstract Mathematics
The Square of an Odd Integer
32 = 9 = 8+1, 52 = 25 = 3x8+1 …… 1312 = 17161 = 2145x8 + 1, ………
Idea 1: prove that n2 – 1 is divisible by 8.
Idea 2: consider (2k+1)2
Idea 0: find counterexample.
Idea 3: Use quotient-remainder theorem.
28Introduction to Abstract Mathematics
Since m is an odd number, m = 2l+1 for some natural number l.
So m2 is an odd number.
Contrapositive Proof
Statement: If m2 is even, then m is evenContrapositive: If m is odd, then m2 is odd.
So m2 = (2l+1)2= (2l)2 + 2(2l) + 1
Proof (the contrapositive):
Proof by contrapositive.
29Introduction to Abstract Mathematics
• Suppose was rational.
• Choose m, n integers without common prime factors (always
possible) such that
• Show that m and n are both even, thus having a common
factor 2,
a contradiction!
n
m2
Theorem: is irrational.2
Proof (by contradiction):
Irrational Number
2
30Introduction to Abstract Mathematics
lm 2so can assume
2 24m l
22 2ln
so n is even.
n
m2
mn2
222 mn
so m is even.
2 22 4n l
Theorem: is irrational.2
Proof (by contradiction): Want to prove both m and n are even.
Proof by contradiction.
Irrational Number
31Introduction to Abstract Mathematics
Infinitude of the Primes
Theorem. There are infinitely many prime numbers.
Claim: if p divides a, then p does not divide a+1.
Let p1, p2, …, pN be all the primes.
Proof by contradiction.
Consider p1p2…pN + 1.
32Introduction to Abstract Mathematics
Floor and Ceiling
If k is an integer, what are x and x + 1/2 ?
Is x + y = x + y? ( what if x = 0.6 and y = 0.7)
For all real numbers x and all integers m, x + m = x + m
For any integer n, n/2 is n/2 for even n and (n–1)/2 for odd n
Def: For any real number x, the floor of x, written x, is the unique integer n such that n x < n + 1.
It is the largest integer not exceeding x ( x).
Def: For any real number x, the ceiling of x, written x, is the unique integer n such that n – 1 < x n. What is n?
33Introduction to Abstract Mathematics
Exercises
Is it true that for all real numbers x and y:
x – y = x - y
x – 1 = x - 1
x + y = x + y
x + 1 = x + 1
For positive integers n and d, n = d * q + r, where d = n / d and r = n – d * n / d with 0 r < d
34Introduction to Abstract Mathematics
Greatest Common Divisors
Given a and b, how to compute gcd(a,b)?
Can try every number, but can we do it more efficiently?
Let’s say a>b.
1. If a=kb, then gcd(a,b)=b, and we are done.
2. Otherwise, by the Division Theorem, a = qb + r for r>0.
35Introduction to Abstract Mathematics
Greatest Common Divisors
Let’s say a>b.
1. If a=kb, then gcd(a,b)=b, and we are done.
2. Otherwise, by the Division Theorem, a = qb + r for r>0.
Euclid: gcd(a,b) = gcd(b,r)!
a=12, b=8 => 12 = 8 + 4 gcd(12,8) = 4
a=21, b=9 => 21 = 2x9 + 3 gcd(21,9) = 3
a=99, b=27 => 99 = 3x27 + 18 gcd(99,27) = 9
gcd(8,4) = 4
gcd(9,3) = 3
gcd(27,18) = 9
36Introduction to Abstract Mathematics
Euclid’s GCD Algorithm
Euclid: gcd(a,b) = gcd(b,r)
gcd(a,b)
if b = 0, then answer = a.
else
write a = qb + r
answer = gcd(b,r)
a = qb + r
37Introduction to Abstract Mathematics
gcd(a,b)
if b = 0, then answer = a.
else
write a = qb + r
answer = gcd(b,r)
Example 1
GCD(102, 70) 102 = 70 + 32
= GCD(70, 32) 70 = 2x32 +
6
= GCD(32, 6) 32 = 5x6 +
2
= GCD(6, 2) 6 = 3x2 + 0
= GCD(2, 0)
Return value: 2.Example 2
GCD(252, 189) 252 = 1x189 + 63
= GCD(189, 63) 189 = 3x63 +
0
= GCD(63, 0)
Return value: 63.
GCD(662, 414) 662 = 1x414 + 248
= GCD(414, 248) 414 = 1x248 +
166
= GCD(248, 166) 248 = 1x166 +
82
= GCD(166, 82) 166 = 2x82 +
2
= GCD(82, 2) 82 = 41x2 + 0
= GCD(2, 0)
Return value: 2.
Example 3
39Introduction to Abstract Mathematics
Practice problems
1. Study the Sections 3.1- 3.4 from your textbook.
2. Be sure that you understand all the examples discussed in class and in textbook.
3. Do the following problems from the textbook:
Exercise 3.1 # 13, 16, 32, 36, 45 Exercise 3.2 # 15, 19, 21, 32, Exercise 3.3 # 13, 16, 25, 26, Exercise 3.4 # 4, 6, 8, 10, 18, 33