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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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last time
Last time
  • What did we talk about last time?
  • Proving universal statements
    • Disproving existential statements
  • Rational numbers
  • Proof formatting
logical warmup
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
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
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
Prove or disprove:
  • The reciprocal of any rational number is a rational number
using existing theorems
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
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
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
prove or disprove1
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
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
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
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
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
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
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
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
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
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
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
  • 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
Next time…
  • Indirect argument
    • Proof by contradiction
  • Irrationality of the square root of 2
  • Infinite number of primes
  • Turn in Assignment 2 by midnight tonight!
  • Keep reading Chapter 4