Download Direct Proofs (continued)

In this lecture
Number Theory
● Rational numbers
● Divisibility
Proofs
● Direct proofs (cont.)
● Common mistakes in proofs
● Disproof by counterexample
1
Common mistakes in proofs
• Arguing from examples
• Using same letter
to mean two different things
• Jumping to a conclusion
(without adequate reasons)
2
Disproof by counterexample
 To disprove statement of the form
“xD if P(x) then Q(x)”,
find a value of x for which
● P(x) is true and
● Q(x) is false.
 Ex: For any prime number a,
a2-1 is even integer.
Counterexample: a=2.
3
Rational Numbers
Definition:
r is rational iff
 integers a and b such that
r=a/b and b≠0.
Examples:
5/6, -178/123, 36, 0, 0.256256256…
Theorem: Every integer
is a rational number.
4
Properties of Rational Numbers
• Theorem: The sum of two rational
numbers is rational.
• Proof:
Suppose r and s are rational numbers.
Then r=a/b and s=c/d for some integers
a,b,c,d s.t. b≠0, d≠0. (by definition)
a
c
r

s


So
(by substitution)
b
d
ad  bc
(by basic algebra)

bd
Let p=ad+bc and q=bd.
Then r+s=p/q where p,q Z and q≠0.
Thus, r+s is rational by definition.
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5
Types of Mathematical
Statements
Theorems: Very important statements that
have many and varied consequences.
Propositions: Less important and
consequential.
Corollaries: The truth can be deduced
almost immediately
from other statements.
Lemmas: Don’t have much intrinsic interest
but help to prove other theorems.
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Divisibility
• Definition: For n,d Z and d≠0 we say that
n is divisible by d
iff n=d·k for some k Z .
• Alternative ways to say:
n is a multiple of d , d is a factor of n ,
d is a divisor of n , d divides n .
• Notation: d | n .
• Examples: 6|48, 5|5, -4|8, 7|0, 1|9 .
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Properties of Divisibility
• For xZ, 1|x .
• For xZ s.t. x≠0, x|0 .
• An integer x>1 is prime
iff its only positive divisors are 1 and x .
• For a,b,cZ, if a|b and a|c then a|(b+c) .
• Transitivity: For a,b,cZ,
if a|b and b|c then a|c .
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Divisibility by a prime
 Theorem: Any integer n>1
is divisible by a prime number.
 Sketch of proof:
Division into cases:
● If n is prime then we are done (since n | n).
● If n is composite
then n=r1·s1 where r1,s1 Z and 1<r1<n,1<s1<n.
(by definition of composite number)
(Further) division into cases:
♦If r1 is prime then we are done (since r1 |n).
♦ If r1 is composite
then r1=r2·s2 where r2,s2 Z and 1<r2<r1,1<s2<r1.
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Divisibility by a prime
 Sketch of proof (cont.):
Since r1|n and r2|r1 then r2 |n (by transitivity).
Continuing the division into cases,
we will get a sequence of integers
r1 , r2 , r3 ,…, rk
such that 1< rk< rk-1<…< r2< r1<n ;
rp |n for each p=1,2,…,k ;
rk is prime.
Thus, r is a prime that divides n.
k
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10
Unique Factorization Theorem
• Theorem:
For  integer n>1,
 positive integer k,
distinct prime numbers p1 , p2 ,, pk ,
positive integers e1 , e2 ,, ek
ek
e1
e2
s.t. n  p1  p2    pk ,
and this factorization is unique.
6
2
3
• Example: 72,000 = 2  3  5
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