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Review Mathematical Induction.ppt

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					             Review: Mathematical Induction


    Use induction to prove that the sum
     of the first n odd integers is n2.

    Base case (n=1): the sum of the first 1 odd integer
      is 12. Yes, 1 = 12.

    Assume P(k): the sum of the first k odd ints is k2.
      1 + 3 + … + (2k - 1) = k2

    Prove that 1 + 3 + … + (2k - 1) + (2k + 1) = (k+1)2

           1 + 3 + … + (2k-1) + (2k+1) = k2 + (2k + 1)
                                       = (k+1)2
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    Mathematical Induction -                      a cool example

          Deficient Tiling
          A 2n x 2n sized grid is deficient
           if all but one cell is tiled.
                                             2n




                      2n



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    Mathematical Induction -                     a cool example



         • We want to show that all 2n x 2n sized
           deficient grids can be tiled with tiles, called
           triominoes, shaped like:




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    Mathematical Induction -                   a cool example




         • Is it true for all 21 x 21 grids?




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    Mathematical Induction -                 a cool example


     Inductive Hypothesis:
     We can tile any 2k x 2k deficient
       board using our fancy designer
       tiles.

     Use this to prove:
     We can tile any 2k+1 x 2k+1
      deficient board using our fancy
      designer tiles.



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    Mathematical Induction -                           a cool example

                                             2k   2k




                  2k
                                        ?         ?
                                                          2k+1


                  2k
                                       OK!!
                                       (by
                                       IH)
                                                  ?
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    Mathematical Induction -                             a cool example

                                             2k    2k



                                       OK!!       OK!!
                  2k                   (by        (by
                                       IH)        IH)

                                                            2k+1
                                       OK!!       OK!!
                  2k                   (by        (by
                                       IH)        IH)



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    Mathematical Induction -                 a cool example




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   Mathematical Induction -                  why does it work?


      Definition:
      A set S is “well-ordered” if every non
       -empty subset of S has a least
       element.

      Given (we take as an axiom): the set
        of natural numbers (N) is well-
        ordered.
                                                        No.
                                              { x Î Z : x < 0 } has no
      Is the set of integers (Z) well             least element.
        ordered?
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   Mathematical Induction -                     why does it work?



    Is the set of non-negative reals (R)
      well ordered?




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   Mathematical Induction -                                why does it work?



    Proof of Mathematical Induction:

    We prove that (P(0) Ù ("k P(k) ® P(k+1)))
     ® ("n P(n))
    Assume
    • P(0)
    • "k P(k) ® P(k+1)
    • Ø"n P(n)                                  $n ØP(n)




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   Mathematical Induction -                                why does it work?


    Assume
    • P(0)
    • "n P(n) ® P(n+1)
    • Ø"n P(n)                                  $n ØP(n)


    Let S = { n : ØP(n) }


  What do we know?
  -P(k) is false because it’s in S.
  -k ¹ 0 because P(0) is true.
  -P(k-1) is true because P(k) is the least element
     in S.
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              Strong Mathematical Induction


    If
      P(0) and
      "n³0 (P(0) Ù P(1) Ù … Ù P(n)) ® P(n+1)

    Then
                                            In our proofs, to show P(k+1), our
      "n³0 P(n)                                 inductive hypothesis assumes
                                              that ALL of P(0), P(1), … P(k) are
                                              true, so we can use ANY of them
                                                   to make the inference.




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                               Game with Matches
     • Two players take turns removing any
       number of matches from one of two piles
       of matches. The player who removes the
       last match wins




     • Show that if two piles contain the same
       number of matches initially, then the
       second player is guaranteed a win

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                   Strategy for Second Player
     • Let P(n) denote the statement “the second
       player wins when they are initially n matches in
       each pile”

     • Basis step: P(1) is true, because only 1 match in
       each pile, first player must remove one match
       from one pile. Second player removes other
       match and wins

     • Inductive step: suppose P(j) is True for all j
       1<=j <= k.

     • Prove that P(k+1) is true, that is the second
       player wins when each piles contains k+1
       matches

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                   Strategy for Second Player
     • Suppose that the first player removes r
       matches from one pile, leaving k+1 –r
       matches there

     • By removing the same number of matches
       from the other pile the second player
       creates the situation of two piles with
       k+1-r matches in each. Apply the
       inductive hypothesis and the second
       player wins each time.


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                         Postage Stamp Example
     • Prove that every amount of postage of 12
       cents or more can be formed using just 4
       -cent and 5-cent stamps

     • P(n) : Postage of n cents can be formed
       using 4-cent and 5-cent stamps

     • All n >= 12, P(n) is true




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                             Postage Stamp Proof
  • Base Case: n = 12, n = 13, n = 14, n = 15
     – We can form postage of 12 cents using 3, 4-cent stamps
     – We can form postage of 13 cents using 2, 4- cent stamps
       and 1 5-cent stamp
     – We can form postage of 14 cents using 1, 4-cent stamp and
       2 5-cent stamps
     – We can form postage of 15 cents using 3, 5-cent stamps
  • Induction Step
     – Let n >= 15
     – Assume P(k) is true for 12 <= k <= n, that is postage of k
       cents can be formed with 4-cent and 5-cent stamps
       (Inductive Hypothesis)
     – Prove P(n+1)
     – To form postage of n +1 cents, use the stamps that form
       postage of n-3 cents (from I.H) with a 4-cent stamp




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                               Recursive Definitions

    We completely understand the function f(n)
     = n!, right?

    As a reminder, here’s the definition:
       n! = 1 · 2 · 3 · … · (n-1) · n, n ³ 1
         But equivalently, we could define it like this:




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                             Recursive Definitions


    Another VERY common example:

    Fibonacci Numbers




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