Lets proceed to

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Lets proceed to

• Proofs
• Text Section 1.5

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Mathematical Reasoning

• We need mathematical reasoning to
• determine whether a mathematical argument is
  correct or incorrect and
• construct mathematical arguments.
• Mathematical reasoning is not only important for
  conducting proofs and program verification, but
  also for artificial intelligence systems (drawing
  inferences).

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Terminology

• An axiom is a basic assumption about mathematical
  structures that needs no proof.
• We can use a proof to demonstrate that a
  particular statement is true. A proof consists of
  a sequence of statements that form an argument.
• The steps that connect the statements in such a
  sequence are the rules of inference.
• Cases of incorrect reasoning are called
  fallacies.
• A theorem is a statement that can be shown to be
  true.

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Terminology

• A lemma is a simple theorem used as an
  intermediate result in the proof of another
  theorem.
• A corollary is a proposition that follows
  directly from a theorem that has been proved.
• A conjecture is a statement whose truth value is
  unknown. Once it is proven, it becomes a theorem.

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Rules of Inference
p p ? q ____ ? q
p?q q?r _____ ? p?r
Modusponens
Hypothetical syllogism
?q p?q _____ ? ?p
p?q ?p _____ ? q
Modus tollens
Disjunctive syllogism
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Rules of Inference
Addition
Conjunction
p _____ ? p?q
p q _____ ? p?q
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Writing Proofs

• Write the statement to be proved.
• Clearly mark the beginning of your proof with the
  word Proof.
• Make your proof self-contained
• Identify each variable used in the body of the
  proof
• Introduce only necessary variables and notation
• Use Lemmas to show significant but related ideas.
• Write proofs in complete (English) sentences.

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Proof Methods

• There are various techniques which are used to
  prove theorems. Choosing which technique to use
  is determined by the type of theorem to be proved
  and your experience.

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Direct Proof

• Direct proof
• An implication p?q can be proved by showing that
  if p is true, then q is also true.
• We use the hypothesis, p, and all known axioms
  and definitions to form a valid argument the
  shows that q must also be true.

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Direct Proof

• Most theorems are of the form
• "x ÎU if P(x), then Q(x).
• If U is a finite set, we can just exhaust over
  each element n to verify that Q(n) holds.
• Example Prove all n Î 4, 6, 8, 10, 12 can be
  written as the sum of two primes.
• Proof 4 2 2 6 3 3
• 8 3 5 10 3 7
• 12 5 7.

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Direct Proof

• When it is not feasible to exhaust over each
  element of the domain, we turn to the method of
  generalizing from the generic particular using
  the rule of Universal Generalization
• To show that every element of a universe
  satisfies a certain property, suppose x is a
  particular, but arbitrarily chosen element of the
  universe, and show that x satisfies the property.
• This is the strategy we employ in the method of
  direct proof.

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Direct Proof Example

• Example Give a direct proof of the theorem If
  n is odd, then n2 is odd.
• This theorem could be formally stated as
• For all integers, n, if n is odd, then n2 is
  odd
• For all integers, if an integer is odd, then the
  square of the integer is odd
• For all odd integers, n, n2 is odd
• All of which are of the form "x ÎU if P(x), then
  Q(x).

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If n is odd, n2 is odd

• The idea is to assume that the hypothesis, p, (n
  is an odd integer) is true for an arbitrary, but
  particular member of the universe. Then use
  rules of inference and known theorems, axioms and
  definitions to show that q must also be true (n2
  is odd).
• This is an application of universal
  generalization.

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If n is odd, n2 is odd

• Proof Let n be an odd integer.
• Then n 2k 1, where k is an integer.To
  show n2 2m 1, where m is an integer.
• Since n 2k 1
• Then n2 (2k 1)2.
• 4k2 4k 1
• 2(2k2 2k) 1
• 2m 1, where m 2k2 2k
• Since n2 can be written in this form, it is
  odd.
• Q.E.D.

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Product of Rationals

• Prove The product of two rational numbers is
  rational.
• Proof
• Let a and b be arbitrary rational numbers.Then
  by definition, there exist integers w, x, y and z
  such that a x/y and b w/z and y ? 0 and z ?
  0.
• The product ab (x/y)(w/z) xw/yz. Since x
  and w are integers, xw is an integer. Since y and
  z are integers and y ? 0 and z ? 0, yz is an
  integer and yz ? 0.Therefore, by definition, ab
  is a rational number.
• Q. E. D.

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Direct Proof Exercises

• Prove the following using the direct proof
  technique.
• The difference of any two rational numbers is a
  rational number.
• If n is an odd integer, then n2 n is even.
• The sum of two odd integers is even.

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Indirect proof

• An implication p?q is logically equivalent to its
  contra-positive ?q ? ?p. Therefore, we can prove
  p?q indirectly by proving its contra-positive
  (with a direct proof). We show that whenever q
  is false, then p is also false.
• Example Give an indirect proof of the theorem
  If 3n 2 is odd, then n is odd.
• Idea Assume that the conclusion of this
  implication is false (n is even). Then use rules
  of inference and known theorems to show that p
  must also be false (3n 2 is even).

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If 3n 2 is odd, n is odd

• Proof (of the contra-positive)
• Let n be any even integer.
• Then n 2k, for some integer, k.
• It follows that 3n 2 3(2k) 2
• 6k 2
• 2(3k 1)
• Therefore, 3n 2 is even.
• Q. E. D.
• We have shown that the contra-positive of the
  implication is true, so the implication itself is
  also true (If 3n 2 is odd, then n is odd).

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n3 5

• Prove If n is any integer and n3 5 is odd,
  then n is even. We prove the contra-positive (if
  n is any integer and n is odd, then n3 5 is
  even) as an example of indirect proof.
• Proof
• Let n be any odd integer.Then n 2k 1 for
  some integer, kAnd n3 5 (2k1)3 5
• (8k3 4K2 4k 1) 5
• 8k3 4k2 4k 6
• 2(4k3 2k2 3k 3)
• n3 5 can therefore written as 2 times some
  integer and by definition is even.
• Q. E. D

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Indirect Proof Exercises

• (Note that a direct proof is also possible)
• If the product of two positive real numbers is
  greater than 100, then at least one of the
  numbers is greater than 10.
• Prove that the sum of two odd integers is even.

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The rule of contradiction

• Although not listed as a rule of inference by our
  text, the following rule is also a valid argument
• p ? False
• ? ? p
• That is, if p implies a contradiction, then p
  must be false and hence ? p must be true.

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Proof by contradiction

• We can use the rule of contradiction to prove
  theorems. This technique is often applied to
  those theorem that assert that some property is
  true. We assume that the property is false and
  find that that assumption leads to some
  contradiction.
• Example Prove ?2 is irrational using proof by
  contradiction.
• Idea Assume ?2 is rational and look for a
  contradiction.

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?2 is irrational

• Prove ?2 is irrational (p)
• Proof
• Assume ?2 is rational (? p)
• Then there exists integers a, b such that 2
  a / b where a and b have no common factors.
• Since ?2 a / b
• 2 a2 / b2
• 2b2 a2

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?2 is irrational

• Therefore a2 is even and a is even and a 2c
  for some integer c.
• Hence,
• 2b2 a2 (2c)2 4c2
• b2 2c2
• Therefore b2 is even and b is even.

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?2 is irrational

• This is the contradiction.
• Assuming ?2 is rational (? p)
• implies both
• a and b have no common factors ( r ) and
• both a and b are even and therefore have a common
  factor(?r)
• Therefore ?2 is rational (? p) must be false
  (it implies both r and ?r, a contradiction) and
  hence ?2 is irrational (p) must be true.
• Q.E.D.

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Infinite Primes

• Prove There are infinitely many primes.
• Proof
• In class.

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Contradiction Exercise

• Prove each of the following by contradiction.
• If n3 5 is odd, then n is odd
• For all integers a, b, and c, if bc is not
  divisible by a, then b is not divisible by a.
• For all integers m and n, if m n is even, then
  both m and n are even or both m and n are odd.

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Proof by Cases

• Sometimes the implication p ? q can be restated
  as (p1 ? p2 ? p3 ? ? pn) ? q.
• In that case we use the tautology
• ((p1 ? p2 ? p3 ? ?pn) ? q ?
• (p1 ? q) ? (p2 ? q) ? (p3? q) ? (pn ? q)
• Each pi is proven as a case.
• Some common cases in mathematics
• All integers are either odd or even
• All integers are either positive, negative or
  zero
• For all real numbers x and y, x gt y, x y or x
  lt y

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xy xy

• Prove xy xy where x and y are real
  numbers.
• Idea Observing thatn n when n ? 0 and n
  -n when n lt 0 leads us to consider categorizing
  the real numbers x, y as ? 0 or lt 0 which leads
  to 4 cases.

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xy xy

• Proof
• Case 1 Let x ? 0 and y ? 0
• then x x and y y and xy gt 0 then xy
  xy xy
• Case 2 Let x ? 0 and y lt 0
• then x x and y -y and xy lt 0
• Therefore xy -xy (since xy lt 0)
• x(-y)
• xy

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xy xy

• Case 3 Let x lt 0 and y ? 0
• then x -x and y y and xy lt 0
• then xy -xy (-x)y xy.
• Case 4 Let x lt 0 and y lt 0
• then x -x and y -y and xy gt 0
• then xy xy (-x)(-y) xy.
• Q. E. D.

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min(x,y) max(x, y)

• Prove If x and y are real numbers, then max(x,
  y) min(x, y) x y.
• Proof (by cases)
• Case 1 x ? y If x ? y, then max(x, y) x and
  min(x, y) y and therefore max(x,y) min(x, y)
  x y
• Case 2 x lt y
• If x lt y then max(x, y) y and min(x,y) x
• Therefore
• max(x,y) min(x,y) y x x y
• Q. E. D.

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Case Examples

• Prove the following by using proof by cases
• Any two consecutive integers have opposite parity
  (one is odd and one is even).
• Any integer, n, can be written as n 3k, n 3k
  1 or n 3k 2. Using this fact, prove that
  the product of any three consecutive integers is
  divisible by 3.

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Proving Biconditionals

• To prove the biconditional p ? q, (read p if and
  only if q) we use the tautology (p? q) ? (p
  ?q) ? (q ? p).
• That is, to prove p ? q, prove both
• p ?q and q ? p.
• Example Prove the integer n is odd if and only
  if n2 is odd.
• Idea Prove if n is odd, n2 is odd and prove
  if n2 is odd, n is odd.
• This is left as an exercise for the student.

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Bi-Conditional Exercises

• Prove that m2 n2 if and only if m n or m
  -n. Hint m2 n2 (m n)(m n)
• Prove that if n is a positive integer, then n is
  odd if and only if 5n 6 is odd.

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Existence Proofs

• Many theorem are simply statements that objects
  of a certain type or with certain properties
  exist. These theorems are of the form
• ?x P( x )

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Uniqueness Proofs

• Some theorems assert that only one object with a
  particular product exists.
• To prove these theorem, prove the object exists
  and then show its unique.
• Proving an object is unique is usually
  accomplished with proof by contradiction.
• See example 28, page 70 of the text.

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