Discrete Mathematics CS 2610

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Discrete Mathematics CS 2610
March 26, 2009
Skip structural induction generalized
induction Skip section 4.5
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Well-Ordering Property

• Every non-empty set of non-negative integers has
  a minimum (smallest) element
• The well-ordering property is the foundation of
  Mathematical induction

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Recursion

• Recursion means defining an object in terms of
• itself
• part of itself
• versions of itself
• An object can be
• Sequence
• Function
• Set
• Algorithm

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In Nature

• Fractals are self-similar structures, most of
  them defined recursively

A fractal coastline in northern Portugal
Romanesque Broccoli
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Recursively Defined Sequence

• Def. A sequence is defined recursively whenever
  some initial terms are specified and later terms
  are defined in terms of earlier terms.
• Arithmetic Series
• a01, r3
• anan-1r, ngt0 yields 1, 4, 7, 10, 13,
• Geometric Series
• a03, r2
• anan-1r, ngt0 yields 3, 6, 12, 24, 48,
• an 2n yields 1, 2, 4, 8, 16, 32,
  
• or an1 2an where n gt 0 and a0 1

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Recursively Defined Sequence

• In a recursively defined sequence
• Base or Initial Conditions
• The first term(s) of the sequence are defined
• Recursion or Recursive Step
• The nth term is defined in terms of previous
  terms
• The formula to express the nth term is called a
  recurrence formula
• Arithmetic Series
• Base a01, r3
• Recursion anan-1r, n gt 0
• Geometric Series
• Base a03, r2
• Recursion anan-1r, n gt 0

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Recursively Defined Sequence

• Be sure that the recursive definition of the
  sequence produces a well-defined sequence in
  which all the terms 0,1,2 are covered by the
  definition
• Example
• Base a0 1
• Recursion an 3an-2 - an-1
• What is a1? Cant tell so this is no good.

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Fibonacci Sequence

• Fibonnaci Sequence
• Non-recursive (closed form) Definition
• Recursive Definition
• Base Cases f00, f11
• Recurrence fn fn-1 fn-2 for n gt 1

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Fibonacci Sequence

• Theorem fn lt 2n.
• Proof (By strong induction.)
• Base cases
• f0 0 lt 20 1f1 1 lt 21 2
• Inductive step
• Inductive Hypothesis Assume ?j, 1 lt j ? k, fj
  lt 2j.
• i.e., f2 lt 22, f3 lt 23, , fk lt 2k
• show that fk1 lt 2k1
• fk fk-1 lt 22k 2k 2k
• fk lt 2k from ind hyp and fk-1 fk lt 2k
  so
• fk1 lt 2k1

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Fibonacci Sequence

• Theorem. ? n 3, fn gt an-2, where a (1?5)/2.
• Proof.
• First note that
• a2 a - 1 (1 2?5 5)/4 - (1?5)/2 1
• 6/4 2?5/4 1/2 - ?5/2 1
• ?5/2 - ?5/2 3/2 1/2 1
• 0 (recall quadratic formula)
• Therefore, a2 a 1
• Take it away!

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Recursively Defined Function

• A function f(n) with domain N or a subset of N is
  defined recursively, when f(n) is defined in
  terms of the previous functions of m lt n
• Basis f(0) 1
• Recursion
• Define f(n) from f defined on smaller terms
• Example
• Let f N -gt N defined recursively as
• Basis f(0) 1
• Recursion f(n 1) (n 1) f(n).
• What are the values of the following?
• f(1) 1 f(2) 2 f(3) 6 f(4) 24
• What does this function compute?

n!
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Recursively Defined Function

• Be sure that the recursive definition produces a
  well-defined function, i.e., every element in the
  domain has an image under f
• Example
• Base f(1) 1
• Recursion f(n) 1 f(?n/2? ), n 1
• Is this function correctly defined ?

Ill-defined f(1) is 1 but is not well-defined
according to the recursion and f(0) is not
defined.
f(0) ? f(6) ?
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Recursively Defined Set

• An infinite set S may be defined recursively, by
  giving
• Basis Step A finite set of base elements
• Recursive Step a rule for forming new elements
  in the set from those already in the set
• Exclusion Rule specifies that the set only
  contains those elements specified in the basis
  step or those generated by the recursive step
• Example
• Let S be defined as follows
• Basis Step 1 ? S
• Recursive Step if n ? S then 2n ? S

S 2k k ? N
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Recursively Defined Set

• Example
• Basis Step 0 ?? S
• Recursive Step if m ? S then m 1 ? S
• Exclusion Rule No other numbers are in S.
• What is S?

Example Basis Step 1 ?? S Recursive Step if
m ? S then -1m ? S Exclusion Rule No other
numbers are in S What is S?
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Set of Strings

• Def.An alphabet ? is a finite non-empty set of
  symbols (e.g., ? 0, 1 )
• Def.A String over an alphabet ? is a finite
  sequence of symbols from ? (e.g., 11010 )
• The set ? of strings over ? can be defined as
• Basis Step ? ? S where ? is the empty string
  containing no symbols
• Recursive Step if w ? S and x ? S then wx ? S
  

Is ? countable or uncountable ?
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Recursive Definition on Strings

• Concatenation (combining two strings)
• Basis Step if w ? S then w? w, where ?
  is the empty string containing no symbols.
• Recursive Step if w1 ? S, w2 ? S and x ? S
  then
• w1(w2 x) ? S (same as (w1 w2) x
  ? S)
• Example
• Sa, b
• Let w1aba, w2a and xb then abaab ? S

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Recursive Definition on Strings

• Length
• Basis Step ? 0
• Recursive Step if w ? S and x ? S then
• wx w 1
• Example
• S a, b
• aba (ab)a ab 1
• ab (a)b a 1 so aba a 2
• a (?)a ? 1 so aba ? 3
• ? 0 so aba 3

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Recursive Function on Strings

• The reversal of a string w, wR, consists of the
  string in reverse order. Give a recursive
  definition of the reversal of a string.
• Example
• w abacd, wR dcaba
• Basis Step if w ? then wR ?
• Recursive Step if w vx where x ? S, v ? S
• then wR xvR

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Well-Formed Formulas

• p,q,r, represent proposition variables T, F and
  the set of logical operators ?, ?, ?, ? , ?
  
• Basis Step
• T, F and p where p is a propositional variable
  is well defined (i.e., a wff)
• Recursive Step If E and G are wff then
• (?E), (E ? G), (E ? G), (E ? G), (E ? G) are wff
• Examples
• (?p), (p ? q), ((?p) ? q) , ((?p) ? q) are wff

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Recursive Structures

• The set of rooted trees, where a rooted tree
  consists of a set of vertices containing a
  distinguished vertex called the root, and edges
  connecting these vertices, can be defined
  recursively by these steps
• Basis Step A single vertex r is a rooted tree.
• Recursive Step Suppose that T1, T2, , Tn are
  disjoint rooted trees with roots r1, r2, rn,
  respectively. Then the graph formed by starting
  with a root r, which is not in any of the rooted
  trees T1, T2, Tn, and adding an edge from r to
  each of the vertices r1, r2, rn is also a
  rooted tree.

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Recursive Structures

• The set of extended binary trees can be defined
  recursively by these steps
• Basis Step The empty set is an extended binary
  tree.
• Recursive Step If T1 and T2 are disjoint
  extended binary trees, then there is an extended
  binary tree, denoted by T1 T2, consisting of a
  root r together with edges connecting the root to
  each of the roots of the left subtree T1 and the
  right subtree T2 when these trees are nonempty.

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Recursive Structures

• The set of full binary trees can be defined
  recursively by these steps
• Basis Step There is a full binary tree
  consisting only of a single vertex r.
• Recursive Step If T1 and T2 are disjoint full
  binary trees, there is a full binary tree,
  denoted by T1 T2, consisting of a root r
  together with edges connecting the root to each
  of the roots of the left subtree T1 and the right
  subtree T2.

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Counting (now in chapter 5)

• The basic counting principles are the product
  rule and sum rule.
• Product Rule Suppose that a procedure can be
  broken down into a sequence of two tasks. If
  there are n ways to do the first task and for
  each of these ways of doing the first task, there
  are m ways to do the second task, then there are
  nm ways to do the procedure.
• Sum Rule If a task can be done either in one of
  n ways or in one of m ways, where none of the set
  of n ways is the same as any of the set of m
  ways, then there are n m ways to do the task.

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Counting

• Product Rule Examples.
• Bill and Ted move into a new house with 12
  bedrooms. How many ways can we assign rooms to
  them?
• Stadium seats are labeled with a letter and a
  two-digit number (00 99). Whats the maximum
  number of seats in the stadium?
• How many license plates can we make using three
  letters followed by three digits?
• .

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Counting

• Sum Rule Examples.
• IHOP offers 16 breakfast items, 22 lunch items,
  and 31 dinner items (all unique). How many
  possible items do we have to choose from?
• How many 8-bit bit strings begin with 1 or end
  with 00?
• - begin with 1 27
• - end with 00 26
• - oops, some have been double counted how many?
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• So, 128 64 32 160 ways
• (principle of inclusion-exclusion)

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Counting

• How many 4-bit bit strings are there that do not
  have two consecutive 1s?
• How many ways can a playoff occur between two
  teams where the winner must win 3 out of 5 games.

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Counting

• The Pigeonhole Principle If k is a positive
  integer and k1 or more objects are placed in k
  boxes, then there is at least one box containing
  two or more of the objects. (prove BWOC)
• Of 367 people, at least two have the same birth
  day.
• For every integer n there is a multiple of n that
  has only 0s and 1s in its decimal expansion.

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Counting

• For every integer n there is a multiple of n that
  has only 0s and 1s in its decimal expansion.
• Let n be a positive integer. Consider the n1
  integers 1, 11, 111, , 111 where the last
  integer is the integer with n1 1s. There are n
  possible remainders when an integer is divided by
  n. Since there are n1 integers in the list, by
  the php there must be at least two with the same
  remainder when divided by n. The larger integer
  minus the smaller integer is a multiple of n (how
  do we know?), which has a decimal expansion
  consisting entirely of 0s and 1s.