Mathematical Induction Recursive Definitions

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Mathematical InductionRecursive Definitions
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Learning Objectives

• What is mathematical induction.
• Examples of mathematical induction.
• The second principle of mathematical induction.
• What is a recursive definition of a set.
• What is the recursive definition of a function.
• Examples.

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Mathematical Induction

• A very special rule of inference!
• Definition A set S is well ordered if every
  subset has a least element.
• Note 0, 1 is not well ordered since (0,1
  does not have a least element.
• Examples
• N is well ordered (under the ? relation)
• Any countably infinite set can be well ordered.
• Z can be well ordered but it is not well ordered
  under the ? relation (Z has no smallest element).
• The set of finite strings over an alphabet using
  lexicographic ordering is well ordered.

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Mathematical Induction

• Let P(x) be a predicate over a well ordered set
  S. The problem is to prove ? xP(x) .The rule of
  inference called
• The (first) principle of Mathematical Induction
  can sometimes be used to establish the
  universally quantified assertion.
• In the case that S N, the natural numbers, the
  principle has the following form.
• P(0)
• P(n) --gt P(n 1)
• ? ?xP(x)

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Mathematical Induction

• The hypotheses are
• H1 P(0) and
• H2 P(n) --gt P(n 1) for n arbitrary.
• H1 is called The Basis Step.
• H2 is called The Induction (Inductive) Step
• _____________________
• We first prove that the predicate is true for
  the smallest element of the set S (0 if S N).
• We then show that it is true for an element x (n
  if S N) implies it is true for the next
  element in the set (n 1 if S N).

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Mathematical Induction

• Then
• knowing it is true for the first element means
  it must be true for the element following the
  first or the second element
• knowing it is true for the second element
  implies it is true for the third
• and so forth.
• Therefore, induction is equivalent to modus
  ponens applied a countable number of times!!

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Mathematical Induction

• It is like a row of dominosIf the nth domino
  falls over the (n1)th, it must fall over, so
  pushing the first one down means all must fall
  down.
• To prove H2 we normally use a Direct Proof .
• Assuming P(n) to be true for arbitrary n is
  called the Induction (Inductive) Hypothesis.

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Mathematical Induction

• An Example Prove that the sum of integers from 1
  to N is N(N1) / 2
• 1) Base case, try P(0), P(1).
• 2) Induction case. Assume it holds for values
  up to n, and prove for n 1.
• 3) Requires understanding how the hypothesis for
  n and n 1 are linked, and reducing the n 1case
  to the size n situation.
• Assume that the sum of values from 1 to n is
  n(n1)/2.
• Ask what is the sum of the values from 1 to n
  1.
• But this can be written as (12 n) (n1).
• Our induction hypothesis tell us that we can
  substitute n ( n 1)/2 to the first term n(n1)
  / 2 (n1) (n1) (n2) / 2

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Mathematical Induction

• Another ExampleProve that the sum of powers of 2
  is one less than the next higher power.
• Verify P(0).
• Suppose P(n) is true for all integers less than
  or equal to n.
• P(n1)

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Mathematical Induction

• Example show that

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Mathematical Induction

• The Second Principle of Mathematical Induction
• The rule of inference becomesThe two rules
  are equivalent but sometimes the second is easier
  to apply. See your text for the classic examples.

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Recursive Definitions

• Recursive or inductive definitions of sets and
  functions on recursively defined sets are
  similar.
• 1. Basis step
• For sets- State the basic building blocks
  (BBB's) of the set.
• For functions- State the values of the function
  on the BBBs.
• 2. Inductive or recursive step
• For sets- Show how to build new things from
  old with some construction rules.
• For functions- Show how to compute the value
  of a function on the new things that can be
  built knowing the value on the old things.

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Recursive Definitions

• 3. Extremal clause
• For sets-If you can't build it with a finite
  number of applications of steps 1. and 2. then
  it isn't in the set.
• For functions- A function defined on a
  recursively defined set does not require an
  extremal clause.

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Recursive Definitions

• Note Your author doesn't mention the extremal
  clause.
• It is a standard part of an inductive definition
  of a set but often ignored (since everybody
  knows it is supposed to be there).
• Also note
• To prove something is in the set you must show
  how to construct it with a finite number of
  applications of the basis and inductive steps.
• To prove something is not in the set is often
  more difficult.

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Recursive Definitions

• Example
• A recursive definition of N
• 1. Basis
• 0 is in N (0 is the BBB).
• 2. Induction
• if n is in N then so is n 1 (how to build new
  objects from old add one to an old object to
  get a new one).
• 3. Extremal clause
• If you can't construct it with a finite number
  of applications of 1. and 2., it isn't in N.

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Recursive Definitions

• Now given the above recursive definition of N we
  can give recursive definitions of functions on N
• 1. f(0) 1 (the initial condition or the value
  of the function on the BBBs).
• 2. f(n 1) (n 1) f(n) (the recurrence
  equation, how to define f on the new objects
  based on its value on old objects)
• f is the factorial function f(n) n!.
• Note how it follows the recursive definition of
  N.

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Recursive Definitions

• Proof of assertions about inductively defined
  objects usually involves a Proof by induction.
• Prove the assertion is true for the BBBs in the
  basis step.
• Prove that if the assertion is true for the old
  objects, it must be true for the new objects you
  can build from the old objects.
• Conclude the assertion must be true for all
  objects.

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Recursive Definitions

• Example
• We define an inductively where n is in N.
• Basis a0 1
• Induction a(n 1) an a
• We want to prove the theorem

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Recursive Definitions

• Proof
• Since the powers of a have been defined
  inductively we must use a proof by induction
  somewhere.
• Get rid of the first quantifier on m by Universal
  Instantiation
• Assume m is arbitrary.
• Now prove the remaining quantified assertion by
  induction

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Recursive Definitions

• by induction
• 1. Basis step Show it holds for n 0.
• The left side becomes ama0 am(1) am
• The right side becomes a m0 am
• Hence, the two sides are equal to the same
  value.
• 2. Induction step The Induction hypothesis
• Assume the assertion is true for n aman am n.
• Now show it is true for n 1.
• The left side becomes ama n1 am(ana)
  (aman)a amn a

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Recursive Definitions

• which follows from
• the inductive step in the definition of a n and
• the induction hypothesis and
• the associativity of multiplication. The right
  side becomes
• a m(n1) a (mn)1 a mn awhich follows
  from
• the inductive definition of the powers of a
• the associativity of addition.

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Recursive Definitions

• Hence, we have shown by induction for an
  arbitrary m that
• Since m was arbitrary, by Universal
  Generalization,
• Q.E.D.

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Recursive Definitions

• Example A recursive definition of the Fibonacci
  sequence
• 1. Basis
• f(0) f(1) 1 (two initial conditions)
• 2. Induction
• f(n 1) f (n) f(n - 1)
• (the recurrence equation).

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Recursive Definitions

• Example

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Recursive Definitions

• Example
• We give an inductive definition of the well
  formed parenthesis strings P
• 1. Basis clause
• ( ) is in P
• 2. Induction clause
• if w is in P then so are ( ) w, (w), and w( )
• 3. Extremal clause

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Recursive Definitions

• Example
• (( )( )) is in P.
• Proof
• 1. ( ) is in P by the basis clause
• 2. ( )( ) must be in P by step 1. and the
  induction clause
• 3. (( ) ( )) must be in P by step 2. and the
  induction clause.
• Q. E. D.

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Recursive Definitions

• Note ))(( ) is not in P. Why? Can you prove it?
• (Hint what can you say about the length of
  strings in P? How can you order the strings in P?)

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Recursive Definitions