Direct Proof and Counterexample I

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Direct Proof and Counterexample I

• Lecture 12
• Section 3.1
• Mon, Jan 31, 2005

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Proving Existential Statements

• Proofs of existential statements are often called
  existence proofs.
• Two types of existence proofs
• Constructive
• Construct the object.
• Prove that it has the necessary properties.
• Non-constructive
• Argue indirectly that the object must exist.

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Example Constructive Proof

• Theorem Given a segment AB, there is a midpoint
  M of AB.
• Proof
• Draw circle A.
• Draw circle B.
• Form ?ABC.
• Bisect ??ACB,
• producing M.

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Justification

• Argue by SAS that triangles ACM and BCM are
  congruent and that AM MB.

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Example Constructive Proof

• Theorem The equation
• x2 7y2 1.
• has a solution in positive integers.
• Proof
• Let x 8 and y 3.
• Then 82 7?32 64 63 1.

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Example Constructive Proof

• Theorem The equation
• x2 67y2 1.
• has a solution in positive integers.
• Proof ?

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Example Constructive Proof

• Theorem If N is a square-free positive integer,
  then the equation
• x2 Ny2 1.
• has a solution in positive integers.

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Example Non-Constructive Proof

• Theorem There exists x ? R such that
• x5 3x 1 0.
• Proof
• Let f(x) x5 3x 1.
• f(1) 1 lt 0 and f(2) 27 gt 0.
• f(x) is a continuous function.
• By the Intermediate Value Theorem, there exists x
  ? 1, 2 such that f(x) 0.

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Disproving Universal Statements

• Construct an instance for which the statement is
  false.
• Also called proof by counterexample.

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Example Proof by Counterexample

• Disprove the conjecture (Fermat) All integers of
  the form 22n 1, for n ? 1, are prime.
• (Dis)proof
• Let n 5.
• 225 1 4294967297.
• 4294967297 641?6700417.

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Example Proof by Counterexample

• Disprove the statement If a function is
  continuous at a point, then it is differentiable
  at that point.
• (Dis)proof
• Let f(x) x and consider the point x 0.
• f(x) is continuous at 0.
• f(x) is not differentiable at 0.

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Disproving Existential Statements

• These can be among the most difficult of all
  proofs.
• Fermats Last Theorem is a famous example
• There is no solution in positive integers of the
  equation
• xn yn zn
• when n gt 2.

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Example Disproving an Existential Statement

• Theorem There is no solution in integers to the
  equation
• x2 y2 101010 2.
• Proof
• A perfect square divided by 4 has remainder 0 or
  1.
• Therefore, x2 y2 divided by 4 has remainder 0,
  1, or 3.

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Example Disproving an Existential Statement

• However, 101010 2 divided by 4 has remainder 2.
• Therefore, x2 y2 ? 101010 2 for any integers
  x and y.