CS322

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# CS322 - PowerPoint PPT Presentation

Week 3 - Friday. CS322. Last time. What did we talk about last time? Proving universal statements Disproving existential statements Rational numbers Proof formatting. Questions?. Logical warmup.

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Presentation Transcript
Week 3 - Friday

### CS322

Last time
• What did we talk about last time?
• Proving universal statements
• Disproving existential statements
• Rational numbers
• Proof formatting
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?
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

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:
• The sum, product, and difference of any two even integers is even
• The sum and difference of any two odd integers are even
• The product of any two odd integers is odd
• The product of any even integer and any odd integer is even
• 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
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"
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:
• 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
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
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
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?
Next time…
• Indirect argument