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CS322 Week 3 - Friday

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Page 1: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

CS322Week 3 - Friday

Page 2: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

Last time

What did we talk about last time? Proving existential statements

Disproving universal statements Proving universal statements

Disproving existential statements Proof formatting

Page 3: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

Questions?

Page 4: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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?

Page 5: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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?

Page 6: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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"

Page 7: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

Disproving an existential statement Flipmode is the squad You just negate the statement and

then prove the resulting universal statement

Page 8: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

Rational NumbersStudent Lecture

Page 9: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

Rational Numbers

Page 10: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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

Page 11: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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?

Page 12: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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

Page 13: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

Prove or disprove:

The reciprocal of any rational number is a rational number

Page 14: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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

Page 15: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

Divisibility

Page 16: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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"

Page 17: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

Examples

Is 37 divisible by 3? Is -7 a factor of 7? Does 6 | 256? Is 0 a multiple of 45?

Page 18: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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?

Page 19: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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

Page 20: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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?

Page 21: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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

Page 22: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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

Page 23: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

Quotient Remainder Theorem and Proof by Cases

Page 24: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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

Page 25: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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

Page 26: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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?

Page 27: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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

Page 28: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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

Page 29: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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

Page 30: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

Floor and Ceiling

Page 31: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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

Page 32: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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

Page 33: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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?

Page 34: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

Upcoming

Page 35: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

Next time…

Indirect argument Proof by contradiction

Irrationality of the square root of 2 Infinite number of primes

Page 36: Week 3 - Friday.  What did we talk about last time?  Proving existential statements  Disproving universal statements  Proving universal statements

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!