Mathematical Induction.

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Mathematical Induction.

• Induction is the most important proof method in
  computer science.

• Suppose you want to prove that the proposition
  P (n) is true
• for all n ?N, where N 0, 1, 2, (an
  infinite set).
• ( i. e. every natural number n has some property
  P)

• You might be able to prove it for 0, 1, 2, But
  you cant check
• one-by-one that all natural numbers have property
  P.

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• The key idea of mathematical induction is start
  with 0 and
• repeatedly add 1.

• Suppose you can show that
• i) 0 has property P and
• ii) whenever you add 1 to a number that has
  property P
• the resulting number also has property P.

This guarantees that as you go through the list
of all natural numbers, every number you
encounter must have property P.
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• The proof method of mathematical induction says
  that you
• just need to do two things
• i) prove it for n 0 (basis case)
• ii) prove if its true for nk, then its
  true for n k1

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Why does this prove it for all n ? N ?
Actually it relies upon the property of integers
N0, 1, 2,
How can we prove ? n ?0 (P(n) ? P (n 1)) ?
Pick arbitrary k ?0 and prove it for n k.
So, in the second step we say take some k ?0
and assume that property P holds for n
k (Induction Hypothesis). Then, (based upon this
assumption) we need to show that the property P
can be implied for n k 1.
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There is a variant if you want to prove it for n
? n0
1) In the base case we explicitly check n n0
2) Pick arbitrary k ? n0 and assume that the
property P holds for n k . (Induction
Hypothesis) Based upon this assumption prove
that the property P can be implied for n k 1.
(Induction Step)
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Example. Prove that for all n?1 n! ? 2n?1.
Proof by induction on n?1.
1). Basis n 1. 1!121?120. So, for the case
n1, we checked that n!?2n?1.
2). Assume that for nk, k is some integer k?1,
inequality holds, i. e. k! ? 2k?1 (IH). We want
to prove that n! ? 2n?1 for n k 1, i. e. we
want to show that (k1)! ? 2(k1)?1 Using
the definition of factorial we have (k1)!
(k1)?k! ? (k1)?2k?1 by IH ? (11)?2k?1
because k?1 2 ?2k?1 2kQED, because
2(k1)?1 2k
By Induction Principle we conclude that n! ? 2n?1
for all n?1
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Induction proof for the summation formula
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(No Transcript)
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In accordance with induction pattern 1) (Base)
P (0) 2) ? n ?0 (P(n) ? P (n 1))
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1) Basis check that the formula is correct for
the case of no terms
2) fix some k ?0 and assume that proposition
holds for n k
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Why induction proofs work? Intuitively, because
we can reach any number n if we start with the
smallest one and add 1 step-by-step. So,
induction principle express this obvious fact.
More formally, induction principle is formulated
in terms of a set A n ?N P (n)
Induction Principle . If A is a subset of N that
satisfies two properties 1) 0?A 2) ?n ? 0 n
?A?(n1)?A, then A N