Proof by Induction and contradiction

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Proof by Induction and contradiction

• Leo Cheung

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The TAs

• Our office is in SHB117, feel free to come if you
  get problems about the course
• Or ask your questions in the newsgroup
• news//news.erg.cuhk.edu.hk/cuhk.cse.csc2110

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The Plan

• Some basic mathematical background
• Mathematical Induction
• Proof by contradiction

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Background - Summation
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Background - Product
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Background - Factorial
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Mathematical Induction - Idea

• You want to prove something (call it f(n)) to be
  true for all natural number ngt1
• Step 1 Base case
• Prove n1 (that is f(1)) is true
• Step 2 Induction
• Prove that if f(k) is true then f(k1) is also
  true, for all kgt1
• Done!

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Why its done?

• All we know (and proved)
• f(1) is true
• if f(k) is true than f(k1) is true for all k gt1

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Example
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And lets try
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Contradiction-Idea

• You want to prove f is true then
• Assume f is false
• By the assumption, you derive some false
  statements
• So the assumption is wrong
• That means f is true

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Examples

• Prove For all integers n, if n2 is odd, then n
  is odd.
• Assume there is a n such that n2 is odd and n is
  even
• n 2k for some k
• n2 (2k)x(2k) 4k2, which is even
• Contradiction!
• So the assumption must be wrong.

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Examples

• Prove There are infinitely many prime numbers.
• Assume not, there are only n primes p1, p2, , pn
• Consider k p1 p2 pn 1
• k is not divisible by any prime pi
• So k is a prime (other than p1, p2, ..., pn)
• Contradiction!

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END
Exercise will be posted on the course webpage