Indirect Argument: Contradiction and Contraposition

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Indirect Argument Contradiction and
Contraposition
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Method of Proof by Contradiction

• Suppose the statement to be proved is false.
• Show that this supposition logically leads to a
  contradiction.
• Conclude that the statement to be proved is true.

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Method of Proof by Contradiction (Ex.)

• Theorem There is no least positive rational
  number.
• Proof Suppose the opposite
• ? least positive rational number x.
• That is, ?x?Q s.t. for ? y?Q, yx. (1)
• Consider the number yx/2.
• xgt0 implies that yx/2gt0. (2)
• x?Q implies that yx/2?Q. (3)
• xgt0 implies that yx/2ltx. (4)
• Based on (2),(3),(4), y?Q and yltx.
• This contradicts (1).
• Thus, the supposition is false and
• there is no least positive rational number.

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Method of Proof by Contraposition

• 1. Express the statement to be proved in the
  form
• ? x?D, if P(x) then Q(x) .
• 2. Rewrite in the contrapositive form
• ? x?D, if Q(x) is false then P(x) is false.
• 3. Prove the contrapositive by a direct proof
• (a) Suppose x is an element of D
• such that Q(x) is false.
• (b) Show that P(x) is false.

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Method of Proof by Contraposition (Example)

• Proposition 1 For any integer n,
• if n2 is even then n is also even.
• Proof The contrapositive is
• For any integer n,
• if n is not even then n2 is not even.
  (1)
• Lets show (1) by direct proof.
• Suppose n is not even.
• Then n is odd. So n2k1 for some k?Z.
• Hence n2 (2k1)24k24k1
• Thus, n2 is not even.

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Comparison of Contradiction and Contrapositive
methods

• Advantage of contradiction method
• Contrapositive method only for universal
  conditional statements.
• Contradiction method is more general.
• Advantage of contrapositive method
• Easier structure after the first step,
  Contrapositive method requires a direct proof.
• Contradiction method normally has more
  complicated structure.

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When to use indirect proof

• Statements starting with There is no.
• (E.g., There is no greatest integer ).
• If the negation of the statement deals with
• sets which are easier to handle with.
• (E.g., is irrational rational numbers are
  more structured and easier to handle with than
  irrational numbers).
• If the infinity of some set to be shown.
• (E.g., The set of prime numbers is infinite ).

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Method of Proof by Contradiction (Ex.)

• Theorem is irrational.
• Proof Assume the opposite is rational.
• Then by definition of rational numbers,
• (1)
• where m and n are integers with no common
  factors.
• ( by dividing m and n by any common factors if
  necessary)
• Squaring both sides of (1),
• Then m22n2 (by basic algebra) (2)

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Method of Proof by Contradiction (Ex.)

• Proof (cont.)
• (2) implies that m2 is even. (by definition)
• Then m is even. (by Proposition 1) (3)
• So m2k for some integer k. (by
  definition) (4)
• By substituting (4) into (2)
• 2n2 m2 (2k)2 4k2 .
• By dividing both sides by 2, n2 2k2 .
• Thus, n2 is even (by definition) and n is even
  (by Prop. 1). (5)
• Based on (3) and (5),
• m and n have a common factor of 2.
• This contradicts (1).

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Infinity of Prime Numbers

• Lemma 1 For any integer a and
• any prime number p,
• if pa then p doesnt divide a1.
• Proof (by contradiction)
• Assume the opposite pa and p(a1).
• Then apn and a1pm for some n,m ?Z.
• So 1(a1)-ap(m-n) which implies that p1.
• But the only integer divisors of 1 are 1 and -1.
• Contradiction.

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Infinity of Prime Numbers

• Theorem The set of prime numbers is infinite.
• Proof (by contradiction) Assume the opposite
• The set of prime numbers is finite.
• Then they can be listed as
• p12, p23, , pn in ascending order.
• Consider M p1 p2pn1.
• pM for some prime number p (1)
• (based on the th-m from lecture 9/22).
• p is one of p1, p2, , pn..
• Thus, p p1 p2pn.. (2)
• By (2) and Lemma 1, p is not a divisor of M.
  (3)
• (3) contradicts (1).