Proof Methods: Part 1

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Proof Methods Part 1

• Sections 3.1-3.6

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Basic Definitions

• Theorem - A statement that can be shown to be
  true
• Proof - A series of statements that form a valid
  argument
• Each statement in the series must be
• Basic fact or definition
• Hypothesis of the theorem you are proving
• Previously proved theorems (lemma or corollary)
• Must end with what you are trying to prove
• More Definitions
• A lemma is a simple theorem used in the proof of
  other theorems
• A corollary is a proposition that can be
  established directly from a theorem that has
  already been proved

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Proof Methods

• Constructive Proof of Existence (Existential)
• Form ?x?D Q(x) is True iff Q(x) is True for at
  least one x?D
• Approach
• A) find an x?D that makes Q(x) True
• B) define a set of directions for finding such an
  x
• Method of Exhaustion (Universal)
• Form ?x?D, if P(x) then Q(x)
• Approach
• A) when D is finite
• B) when finite number of elements satisfy P(x)

Note Any Universal statement can be written in
this form
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Proof Methods (Contd)

• Direct Proof (Universal)
• Form ?x?D, if P(x) then Q(x)
• Approach
• Suppose x is a particular but arbitrarily chosen
  element in D for which P(x) is True Suppose x?D
  and P(x)
• Show conclusion is True by using definitions,
  inference rules, previously established results,
  etc.
• Proof by Counterexample
• Form ?x?D, if P(x) then Q(x)
• Approach
• Find a value of x?D for which P(x) is True and
  Q(x) is false

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Proof Methods (Contd)

• Proof by Contradiction
• Form ?x?D, if P(x) then Q(x)
• Approach
• Suppose the statement to be proved is false
  (negation P?Q)
• Show the supposition leads logically to a
  contradiction
• Conclude the statement to be proved is True
• Proof by Indirect Argument
• Form ?x?D, if P(x) then Q(x)
• Approach
• Rewrite statement in contrapositive form ?x?D, if
  Q(x) then P(x)
• Prove the contraspositive by direct proof
• Suppose x is a particular by arbitrarily chosen
  element in D such that Q(x) is False
• Show P(x) is False

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Basic Number Theory Definitions

• Z Set of all Integers
• Z Set of all Positive Integers
• N Set of Natural Numbers (Z and Zero)
• R Set of Real Numbers
• Addition and multiplication on integers produce
  integers
• (a,b ? Z) ? (ab) ? Z ? (ab) ? Z

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Number Theory Defs (cont.)

• Even/Odd
• n is even is defined as ?k?Z n 2k
• n is odd is defined as ?k?Z n 2k1
• Rational/Irrational
• x is rational is defined as
• ?a,b ?Z x a/b, b?0
• x is irrational is defined as
• ??a,b ?Z x a/b, b?0 or ?a,b ? Z, x ? a/b, b?0
• Prime/Composite
• n is prime is defined as
• if ngt1 and ?r?Z, ?s?Z if nrs then r 1 or s
  1
• n is composite is defined as
• ?r?Z, ??Z where nrs and r ? 1 and s ? 1

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Know Your Definitions!

• Is 0/1 a rational number?
• Yes! 0/1 0
• Is 2/0 a rational number?
• No! r a/b and b?0
• Is 0 an even number?
• Yes! ?k?Z n2k (k0)
• Is 1 a prime number?
• No! n is prime iff ngt1
• Is 2 a prime number?
• Yes! ngt1 and nrs then r1 or s1
• 2gt1 and 221 or 212
• Is 3 a composite number?
• No! nrs and r?1 and s?1
• 3 31 or 313

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Direct Proof

• Prove ?n?Z, if n is even, then n2 is even
• Tabular-style proof
• Suppose n is even hypothesis (particular n?Z
  P(x) is T)
• n2b for some b?Z definition of even
• n2 4b2 algebra
• n2 2(2b2) algebra
• Let k 2b2 be an int mult of integers is an
  integer
• ? n2 is even definition of even

P(x)
Q(x)
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Same Direct Proof

• Prove ?n?Z, if n is even, then n2 is even
• Sentence-style proof
• Assume that n is even. Thus, we know that n 2k
  for some integer k. It follows that n2 4k2
  2(2k2). Therefore n2 is even since it is 2 times
  2k2 which is an integer.

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Structure of a Direct Proof

• Prove ?n?Z, if n is even, then n2 is even
• Proof
• Assume that n is even. Thus, we know that n 2k
  for some integer k. It follows that n2 4k2
  2(2k2). Therefore n2 is even since it is 2 times
  2k2 which is an integer.

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Constructive Proof of Existence

• Prove If a and b are integers is 6a2b even?
• Approach 1
• If 6a2b is even, then it must have the form n2k
• hypothesis
• 6a2b 2k for some k?Z definition of even
• 2(3a2b) 2k algebra
• Let k3a2b be an integer multiplication of
  integers
• ? 6a2b is even
• Approach 2
• Let a1 and b2
• 6(1)2(2) 12
• 12 is even

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More Direct Proofs

• Prove that the sum of two odd integers is an even
  integer
• ?n?Z, if m and n are odd, then mn is even
• Suppose m and n are particular but arbitrarily
  chosen integers such that m and n are
  odd hypothesis
• Show mn is even
• m 2a1 for some a?Z defn of odd
• n 2b1 for some b?Z defn of odd
• mn (2b1)(2a1) addition
• mn 2a2b2 2(ab1) factoring
• Let k ab1 be an integer add. of ints
• ? mn is even defn of even

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Disproof by Counterexample

• Consider the question of disproving a statement
  of the form ?x?D, if P(x), then Q(x)
• Showing this statement is False is the same as
  showing that the negation is True. Negating
  gives ?x?D P(x) ? Q(x)
• Example ?a?R, ?b?R, if a2 b2 then a b
• Method 1 Show the statement is False
• Suppose a and b are particular but arbitrarily
  chosen integers such that a 1 and b -1
• Show P(x) is True (a2 b2) and Q(x) is False (a
  ? b)
• P(x) ? a2 b2 ? (1)2 (-1)2 mult. of ints
• Q(x) ? (1) ? (-1) substitution
• Found an example where P(x) is True and Q(x) is
  False
• ? ?a,b?R, If a2 b2 then a b is False

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Disproof by Counterexample (contd)

• Method 2 Show the negation is True
• Suppose a and b are particular but arbitrarily
  chosen elements of R for which P(x) is True
• Show ?a,b?R a2 b2 ? a ? b
• Let a 2 and b -2
• Then a2 4 and b2 4, so a2 b2
• But 2 ? -2, so a ? b
• ? The negation is True and ?a,b?R a2 b2 ? a ?
  b

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Indirect Argument

• Prove that if x2 is even then x is even
• ?x?Z, if Even(x2) then Even(x)
• Contrapositive
• ?x?Z, if Even(x) then Even(x2) or
• ?x?Z, if Odd(x) then Odd(x2)
• Suppose n is a particular but arbitrarily chosen
  integer such that n is odd
• Show n2 is odd
• n 2k1 defn of odd
• n2 (2k1)2 substitution
• n2 4k2 4k 1 2(2k2 2k) 1 mult
  rearranging
• Let j 2k2 2k be an int mult add of ints
• So n2 2j1
• ? n2 is odd by definition and ?x?Z, if Odd(x)
  then Odd(x2)

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More Direct Proof Examples

• Prove that the product of any two odd integers is
  odd
• ?x?Z, if m and n are odd, then mn is odd
• Suppose m and n are particular but arbitrarily
  chosen integers such that m and n are
  odd hypothesis
• Show mn is odd
• m 2a1 for some a?Z defn of odd
• n 2b1 for some b?Z defn of odd
• mn (2b1)(2a1) addition
• mn 4ab2a2b1 2(2abab)1 factoring
• Let k 2abab be an integer mult, add of ints
• ? mn is odd defn of odd

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Direct Proof Division into Cases

• Prove that either x or x1 is even, for any
  integer x
• ?x?Z, Int(x) ? Int(x1) ? Even
• Suppose n is a particular but arbitrarily chosen
  integer such that it is either even or
  odd hypothesis
• Show n or n1 is even
• Case 1 n is even
• n 2k for some k?Z defn of even
• ? It is true that either n or n1 is even

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Direct Proof Division into Cases (contd)

• Case 2 n is odd
• n 2k1 for some k?Z defn of odd
• n1 (2k1)1 substitution
• n1 2k2 2(k1) rearranging
• Let jk1 be an integer add of ints
• So, n1 2j which is even defn of even
• ? It is true that either n or n1 is even
• Therefore, in either case it is true that either
  n or n1 is even