Docstoc

Mathematical Induction

Document Sample
Mathematical Induction Powered By Docstoc
					             Review: Mathematical Induction


    Use induction to prove that the sum
     of the first n odd integers is n2.
                                                  Prove a base case (n=1)
    Base case (n=1): the sum of the first 1 odd integer
      is 12. Yes, 1 = 12.
                                                     Prove P(k)P(k+1)
    Assume P(k): the sum of the first k odd ints is k2. 1
      + 3 + … + (2k - 1) = k2
                                                           Inductive
                                                           hypothesis
    Prove that 1 + 3 + … + (2k - 1) + (2k + 1) = (k+1)2
                                                           By inductive
            1 + 3 + … + (2k-1) + (2k+1) = k2 + (2k + 1)     hypothesis
                                        = (k+1)2
                                                  By arithmetic
1 Extensible Networking Platform
  - CSE 240 – Logic and Discrete Mathematics                              1
    Mathematical Induction -                        a cool example

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




                       2n



2 Extensible Networking Platform
  - CSE 240 – Logic and Discrete Mathematics                         2
    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:




3 Extensible Networking Platform
  - CSE 240 – Logic and Discrete Mathematics                       3
    Mathematical Induction -                    a cool example




                                                         Yes!


          • Is it true for all 21 x 21 grids?




4 Extensible Networking Platform
  - CSE 240 – Logic and Discrete Mathematics                     4
    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.



5 Extensible Networking Platform
  - CSE 240 – Logic and Discrete Mathematics                    5
    Mathematical Induction -                             a cool example

                                               2k   2k




                   2k
                                          ?         ?
                                                            2k+1


                   2k
                                         OK!!
                                         (by
                                         IH)        ?
6 Extensible Networking Platform
  - CSE 240 – Logic and Discrete Mathematics                              6
    Mathematical Induction -                               a cool example

                                               2k    2k



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

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



7 Extensible Networking Platform
  - CSE 240 – Logic and Discrete Mathematics                                7
    Mathematical Induction -                   a cool example




8 Extensible Networking Platform
  - CSE 240 – Logic and Discrete Mathematics                    8
   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  Z : x < 0 } has no
       Is the set of integers (Z) well              least element.
         ordered?
9 Extensible Networking Platform
  - CSE 240 – Logic and Discrete Mathematics                          9
   Mathematical Induction -                             why does it work?



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

                                                          No.
                                                { x  R : x > 1 } has no
                                                    least element.




 Extensible Networking Platform
10 - CSE 240 – Logic and Discrete Mathematics                              10
   Mathematical Induction -                                why does it work?



    Proof of Mathematical Induction:

    We prove that (P(0)  (k P(k)  P(k+1)))
      (n P(n))
    Assume
                                                               Proof by
    1. P(0)                                                  contradiction.
    2. k P(k)  P(k+1)
    3. n P(n)                                 n P(n)




 Extensible Networking Platform
11 - CSE 240 – Logic and Discrete Mathematics                                 11
   Mathematical Induction -                                         why does it work?


    Assume
    1. P(0)
    2. n P(n)  P(n+1)
    3. n P(n)                                 n P(n)


    Let S = { n : P(n) }                          Since N is well ordered, S has a least
                                                             element. Call it k.

                                                            But by (2), P(k-1)  P(k).
  What do we know?                                          Contradicts P(k-1) true, P(k)
  -P(k) is false because it’s in S.                                    false.
  -k  0 because P(0) is true.
  -P(k-1) is true because P(k) is the least element
     in S.                          Done.
 Extensible Networking Platform
12 - CSE 240 – Logic and Discrete Mathematics                                               12
              Strong Mathematical Induction


    If
      P(0) and
      n0 (P(0)  P(1)  …  P(n))  P(n+1)

    Then
                                            In our proofs, to show P(k+1), our
      n0 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.




 Extensible Networking Platform
13 - CSE 240 – Logic and Discrete Mathematics                                      13
                               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

 Extensible Networking Platform
14 - CSE 240 – Logic and Discrete Mathematics      14
                   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

 Extensible Networking Platform
15 - CSE 240 – Logic and Discrete Mathematics              15
                   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.            How is this
                                                different than
                                                    regular
                                                  induction?
 Extensible Networking Platform
16 - CSE 240 – Logic and Discrete Mathematics                    16
                         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




 Extensible Networking Platform
17 - CSE 240 – Logic and Discrete Mathematics     17
                             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

                                                Why does
                                                this work?

 Extensible Networking Platform
18 - CSE 240 – Logic and Discrete Mathematics                       18
                               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  1
         But equivalently, we could define it like this:
                                                                Inductive
                                                               (Recursive)
    Recursive Case                                              Definition
                                      n  (n  1)! if n  1
                                 n! 
        Base Case                          1 if n  0

 Extensible Networking Platform
19 - CSE 240 – Logic and Discrete Mathematics                                19
                             Recursive Definitions


    Another VERY common example:

    Fibonacci Numbers
                         0                       if n  0
                                                                       Base Cases
                 f (n)  1                       if n  1
                          f (n  1)  f (n  2) if n  1
                                                                     Recursive Case


                                                           éæ    ön æ1 - 5 ön ù
   Is there a non-recursive                             1 ê 1+ 5
  definition for the Fibonacci
                                                f (n) =     ç    ÷ -ç      ÷ú
                                                         5 êè 2 ø è 2 ø ú
                                                           ë                  û
           Numbers?

 Extensible Networking Platform
20 - CSE 240 – Logic and Discrete Mathematics                                          20

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:0
posted:4/20/2013
language:Unknown
pages:20