Proof Methods: Part 2

1
Proof Methods Part 2

• Sections 3.1-3.6

2
More Number Theory Definitions

• Divisibility
• n is divisible by d iff
• ?k?Z ndk
• dn is read d divides n where n and d are
  integers and d ? 0 (note dn ? d/n)
• Other ways to say it
• n is a multiple of d
• d is a factor of n
• d is a divisor of n
• d divides n
• Properties of Divisibility
• Thm 3.3.1 Transitivity
• ?a,b,c ?Z, If a divides b and b divides c, then a
  divides c
• Thm 3.3.2 Divisibility by a Prime
• Any integer ngt1 is divisible by a prime number

3
Transitivity Proof

• Prove that for all integers a, b, and c, If a
  divides b and b divides c, then a divides c
• ?a,b,c ?Z, If ab and bc then ac
• Suppose a, b, and c are ints such that ab and
  bc
• Show ac or c a(int) remember ndk by
  defn
• ab ? b ar for some r?Z defn of div
• bc ? c bs for some s?Z defn of div
• c (ar)s substitution
• c a(rs) associative
• Let krs be an int mult of ints
• So, c ak
• ? a divides c or ac defn of div

4
Proof by Counterexample

• Is it true or false that for all ints a and b, if
  ab and ba then ab?
• ?a,b ?Z, If ab and ba then ab
• Negating gives.
• ?x?Z (ab and ba) ? (a?b)
• Suppose a and b are ints such that ab and ba
• Show a?b
• ab ? b ak for some k?Z defn of div
• ba ? a bl for some l?Z defn of div
• b (bl)k b(lk) substitution assoc
• Since b ? 0, cancel bs giving 1lk (l and k are
  divisors of 1)
• Thus, k and l are both either 1 or -1
• If k l 1, then ba
• If k l -1, then b -a and so a?b
• This means you can find a counterexample by
  taking b -a
• Example, a -2 and b 2 ab 2-2 and ba
  -22 but a?b
• ? The proposed divisibility property is False!

5
More Number Theory Definitions (cont.)

• Quotient-Remainder Theorem
• Given any integer n and positive int d
• ? unique q,r?Z ndq r and 0 r lt d
• Example 11/4
• 11 24 3
• Div/Mod
• n div d int quotient when n is divisible by d
• n mod d int remainder when n is divisible by d
• n div d q n mod d r
• n dq r

6
Even/Odd is a Special Case of Divisibility

• We say that n is divisible by d if ?k?Z ndk
• n is divisible by 2 if ?k?Z n 2k (even)
• The other case is n 2k1 (odd, remainder of 1)
• n is divisible by 3 if ?k?Z n 3k
• The other cases are n 3k1 and n 3k2
• n is divisible by 4 if ?k?Z n 4k
• The other cases are n 4k1, n 4k2, n 4k3
• n is divisible by 5 if ?k?Z n 5k
• The other cases are .

7
Parity of Integers

• How can we prove whether every integer is either
  even or odd?
• By Q-R Theorem we know that n dq r and 0 r
  lt d
• if d 2, then there exists integers q and r such
  that
• n 2q r and 0 r lt 2
• Evaluating the cases gives.
• n 2q 0 n 2q 1
• even parity (n2k) odd parity
  (n2k1)
• Theorem 3.4.2 Any two consecutive integers have
  opposite parity

8
Applying the Q-R Theorem

• Given any integer n, apply the Q-R Theorem to n
  with d 4
• This implies that there exist an integer quotient
  q and remainder r such that
• n 4q r and 0 r lt 4
• Hence,
• n 4q, n 4q1, n 4q2, n 4q3
• Look at Theorem 3.4.3 The square of any odd
  integer has the form 8m1

9
Divisibility Proof

• Prove n2 2 is never divisible by 3 if n is an
  integer
• Discussion What does it mean for a number to be
  divisible by 3? If a is divisible by 3 then ?k?Z
  a3k and the remainder is 0. Other options are
  a remainder of 1 and 2. So, we need to show that
  the remainder when n2 2 is divided by 3 is
  always 1 or 2.
• There are 3 possible cases
• Case 1 n 3k
• Case 2 n 3k 1
• Case 3 n 3k 2

10
n2-2 Proof (cont)

• Suppose n is a particular but arbitrarily chosen
  integer
• Show When n2 2 is divided by 3 the remainder
  is always 1 or 2

Case 1 n 3k for ?k?Z n2-2 (3k)2 - 2
substitution 9k2 - 2 3(3k2) 2 mult
factoring 3(3k2 - 1) 1 rearranging ? The
remainder when dividing by 3 is 1 Case 2 n
3k1 for ?k?Z n2-2 (3k1)2 - 2
substitution 9k2 6k 1 - 2 3(3k2 2k)
1 mult factoring 3(3k2 2k - 1)
2 rearranging ? The remainder when dividing by
3 is 2
11
n2-2 Proof (cont)

• Case 3 n 3k2 for ?k?Z
• n2-2 (3k2)2 - 2 substitution
• 9k2 12k 4 - 2 multiplying
• 3(3k2 4k) 2 rearranging
• ? The remainder when dividing by 3 is 2
• In each case the remainder when dividing n2-2 by
  3 is nonzero. Thus proving the theorem.

12
Unique Factorization Theorem

• Theorem 3.3.3 Any integer n gt 1 is either prime
  or can be written as a product of prime numbers
  in a way that is unique (Fundamental Theorem of
  Arithmetic)
• 1) prime number
• 2) product of prime numbers
• Example
• n 4 22, where 2 is a prime number
• n 7 71, where 7 is a prime number
• n 100 1010 2255, where 2 and 5 are
  prime numbers

13
Standard Factor Form

• Because of UFT, any integer n gt 1 can be written
  in ascending order from left to right
• n p1e1 p2e2 p3e3 . Pkek
• Where
• k is a positive integer
• p1 ? pk are prime numbers
• e1 ? ek are positive integers
• p1 lt p2 lt p3 lt lt pk
• Example
• k 100 1010 2255 2252
• n 3300 10033 425311 2255311 22
  3 52 11

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Even More Number Theory Definitions (cont.)

• Floor/Ceiling
• Floor of x ? ? x ?
• unique integer n such that n x lt n1
• Ceiling of x ? ? x ?
• unique integer n such that n-1 lt x n
• Example
• X 37 / 4 9 ¼
• ? x ? 9 and 9 9 ¼ lt 10
• ? x ? 10 and 9 lt 9 ¼ 10
• Note Floor or Ceiling of an integer is itself!

15
Proof by Counterexample

• Is the following True or False?
• ?x,y?R, ? x y? ? x ? ? y ?
• Method 1
• Suppose x and y are particular but arbitrarily
  chosen real numbers such that x y ½
• Show The statement ? x y? ? x ? ? y ? is
  False
• ? ½ ½ ? ? 1 ? 1 substitution
• ? ½ ? ? ½ ? 0 0 0 substitution
• ? ? x y? ? ? x ? ? y ?
• Method 2
• Rewrite as a negation ? x y? ? ? x ? ? y ?
• Prove negation is True

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Other Floor/Ceiling Theorems

• Theorem 3.5.1 For all Real numbers x and all
  integers m, ? x m ? ? x ? m
• Intuitive since m is an integer and its floor is
  always itself
• Theorem 3.5.2 The Floor of n/2
• n/2 if n is even
• ? n/2 ?
• (n-1)/2 if n is odd
• Intuitive since when n is even, n/2 is an integer
  and the floor of an integer is itself or n/2