week 3 - friday. what did we talk about last time? proving existential statements disproving...

CS322Week 3 - Friday

Last time

What did we talk about last time? Proving existential statements

Disproving universal statements Proving universal statements

Disproving existential statements Proof formatting

Questions?

Logical warmup An anthropologist studying on the Island of Knights

and Knaves is told that an astrologer and a sorcerer are waiting in a tower

When he goes up into the tower, he sees two men in conical hats

One hat is blue and the other is green The anthropologist cannot determine which man is

which by sight, but he needs to find the sorcerer He asks, "Is the sorcerer a Knight?" The man in the blue hat answers, and the

anthropologist is able to deduce which one is which Which one is the sorcerer?

Logical cooldown

After determining that the man in the green hat is the sorcerer, the sorcerer asks a question that has a definite yes or no answer

Nevertheless, the anthropologist, a naturally honest man, could not answer the question, even though he knew the answer

What's the question?

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

Prove or disprove:

The reciprocal of any rational number 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 claim in formal notation Write Proof: State your premises Justify every line you infer from the

premises Write QED after you have demonstrated

the conclusion

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

Quotient Remainder Theorem and Proof by Cases

Proof by cases If you have a premise consisting of clauses

that are ANDed together, you can split them up Each clause can be used in your proof

What if clauses are ORed together? You don't know for sure that they're all true In this situation, you use a proof by cases Assume each of the individual possibilities is

true separately If the proof works out in all possible cases, it

still holds

Proof by cases formatting For a direct proof using cases, follow the

same format that you normally would When you reach your cases, number

them clearly Show that you have proved the

conclusion for each case Finally, after your cases, state that, since

you have shown the conclusion is true for all possible cases, the conclusion must be true in general

Quotient-remainder theorem For any integer n and any positive integer d,

there exist unique integers q and r such that n = dq + r and 0 ≤ r < d

This is a fancy way of saying that you can divide an integer by another integer and get a unique quotient and remainder

We will use div to mean integer division (exactly like / in Java )

We will use mod to mean integer mod (exactly like % in Java)

What are q and r when n = 54 and d = 4?

Even and odd As another way of looking at our earlier

definition of even and odd, we can apply the quotient-remainder theorem with the divisor 2

Thus, for any integer n n = 2q + r and 0 ≤ r < 2

But, the only possible values of r are 0 and 1 So, for any integer n, exactly one of the

following cases must hold: n = 2q + 0 n = 2q + 1

We call even or oddness parity

Consecutive integers have opposite parity Prove that, given any two

consecutive integers, one is even and the other is odd

Hint Divide into two cases: The smaller of the two integers is even The smaller of the two integers is odd

Another proof by cases

Theorem: for all integers n, 3n2 + n + 14 is even

How could we prove this using cases?

Be careful with formatting

Floor and Ceiling

More definitions

For any real number x, the floor of x, written x, is defined as follows: x = the unique integer n such that n ≤

x < n + 1

For any real number x, the ceiling of x, written x, is defined as follows: x = the unique integer n such that n –

1 < x ≤ n

Proofs with floor and ceiling Prove or disprove:

x, y R, x + y = x + y

Prove or disprove: x R, m Z x + m = x + m

Examples Give the floor for each of the following values

25/4 0.999 -2.01

Now, give the ceiling for each of the same values

If there are 4 quarts in a gallon, how many gallon jugs do you need to transport 17 quarts of werewolf blood?

Does this example use floor or ceiling?

Upcoming

Next time…

Indirect argument Proof by contradiction

Irrationality of the square root of 2 Infinite number of primes

Reminders

Turn in Assignment 2 by midnight tonight!

Keep reading Chapter 4 We're leaving for Cargas Systems

essentially right now If you want to come along, just follow

me Transportation is provided!