week 3 - wednesday. what did we talk about last time? basic number theory definitions even and...
TRANSCRIPT
CS322Week 3 - Wednesday
Last time
What did we talk about last time? Basic number theory definitions
Even and odd Prime and composite
Proving existential statements Disproving universal statements
Questions?
Logical warmup
By their nature, manticores lie every Monday, Tuesday and Wednesday and the other days speak the truth
However, unicorns lie on Thursdays, Fridays and Saturdays and speak the truth the other days of the week
A manticore and a unicorn meet and have the following conversation:Manticore: Yesterday I was lying.Unicorn: So was I.
On which day did they meet?
Proving Universal Statements
Method of exhaustion
If the domain is finite, try every possible value.
Example: x Z+, if 4 ≤ x ≤ 20 and x is even, x can
be written as the sum of two prime numbers
Is this familiar to anyone? Goldbach's Conjecture proposes that
this is true for all even integers greater than 2
Generalizing from the generic particular
Pick some specific (but arbitrary) element from the domain
Show that the property holds for that element, just because of that properties that any such element must have
Thus, it must be true for all elements with the property
Example: x Z, if x is even, then x + 1 is odd
Direct proof
Direct proof actually uses the method of generalizing from a generic particular, following these steps:1. Express the statement to be proved in the
form x D, if P(x) then Q(x)2. Suppose that x is some specific (but
arbitrarily chosen) element of D for which P(x) is true
3. Show that the conclusion Q(x) is true by using definitions, other theorems, and the rules for logical inference
Proof formatting
Write Claim: and then the statement of the theorem
Start your proof with Proof: Define everything Write a justification next to every
line Put QED at the end of your proof
Quod erat demonstrandum: "that which was to be shown"
Direct proof example
Prove the sum of any two odd integers is even.
Common mistakes
Arguing from examples Goldbach's conjecture is not a proof, though shown for
numbers up to 1018
Using the same letter to mean two different things m = 2k + 1 and n = 2k + 1
Jumping to a conclusion Skipping steps
Begging the question Assuming the conclusion
Misuse of the word "if A more minor problem, but a premise should not be
invoked with "if"
Disproving an existential statement
Flipmode is the squad You just negate the statement and
then prove the resulting universal statement
Rational NumbersStudent Lecture
Rational Numbers
Another important definition
A real number is rational if and only if it can be expressed at the quotient of two integers with a nonzero denominator
Or, more formally,r is rational a, b Z r = a/b and b 0
Examples
Give an example of a rational number Given an example of an irrational number Is 10/3 rational? Is 0.281 rational? Is 0.12121212… (the digits 12 repeat
forever) rational? Is 0 rational? Is 3/0 rational? Is 3/0 irrational? If m and n are integers and neither is zero, is
(m + n)/mn rational?
Prove the following:
Every integer is a rational number The sum of any two rational numbers
is rational The product of any two rational
numbers is a rational number
Using existing theorems
Math moves forward not by people proving things purely from definitions but also by using existing theorems
"Standing on the shoulders of giants" Given the following:
1. The sum, product, and difference of any two even integers is even
2. The sum and difference of any two odd integers are even3. The product of any two odd integers is odd4. The product of any even integer and any odd integer is even5. The sum of any odd integer and any even integer is odd
Using these theorems, prove that, if a is any even integer and b is any odd integer, then (a2 + b2 + 1)/2 is an integer
Divisibility
Definition of divisibility
If n and d are integers, then n is divisible by d if and only if n = dk for some integer k
Or, more formally: For n, d Z,
n is divisible by d k Z n = dk We also say:
n is a multiple of d d is a factor of n d is a divisor of n d divides n
We use the notation d | n to mean "d divides n"
Examples
Is 37 divisible by 3? Is -7 a factor of 7? Does 6 | 256? Is 0 a multiple of 45?
More on divisors
If a,b Z and a | b, is a ≤ b? Not necessarily!
But, if a,b Z+ and a | b, then a ≤ b Which integers divide 1? If a,b Z, is 3a + 3b divisible by 3? If k,m Z, is 10km divisible by 5?
Transitivity of divisibility
Prove that for all integers a, b, and c, if a | b and b | c, then a | c
Steps: Rewrite the theorem in formal notation Write Proof: State your premises Justify every line you infer from the
premises Write QED after you have demonstrated
the conclusion
Divisibility by a prime
Prove that every integer n > 1 is divisible by a prime
Steps: Rewrite the theorem in formal notation Write Proof: State your premises Justify every line you infer from the premises Write QED after you have demonstrated the
conclusion This one is a little more problematic We aren't really ready to prove something like this
yet
Prove or disprove:
For all integers a and b, if a | b and b | a, then a = b
How could we change this statement so that it is true?
Then, how could we prove it?
Unique factorization theorem For any integer n > 1, there exist a
positive integer k, distinct prime numbers p1, p2, …, pk, and positive integers e1, e2, …, ek such that
And any other expression of n as a product of prime numbers is identical to this except, perhaps, for the order in which the factors are written
kek
eee ppppn ...321321
An application of the unique factorization theorem
Let m be an integer such that 8∙7 ∙6 ∙5 ∙4 ∙3 ∙2 ∙m = 17∙16 ∙15
∙14 ∙13 ∙12 ∙11 ∙10 Does 17 | m? Leave aside for the moment that we
could actually compute m
Upcoming
Next time…
Quotient remainder theorem Proof by cases
Reminders
Keep reading Chapter 4 Work on Assignment 2 for Friday