Induction: One Step At A TimeSlide 2Slide 3Slide 4Slide 5Slide 6Let’s start with dominoesDomino Principle: Line up any number of dominos in a row; knock the first one over and they will all fall.n dominoes numbered 1 to nSlide 10n dominoes numbered 0 to n-1The Natural NumbersPlato: The Domino Principle works for an infinite row of dominoesPlato’s Dominoes One for each natural numberThe Infinite Domino PrincipleMathematical Induction: statements proved instead of dominoes fallenInductive Proof / Reasoning To Prove k, SkSlide 18Inductive Proof / Reasoning To Prove k¸b, SkSlide 20Sn ´ “n =n(n+1)/2” Use induction to prove k¸0, SkSlide 22Slide 23Slide 24Inductive Definition Of FunctionsInductive Definition Recurrence Relation for F(X)Slide 27Slide 28Slide 29Inductive Definition Recurrence Relation for F(X) = 2XInductive Definition Recurrence RelationSlide 32Slide 33Slide 34Inductive Definition Recurrence Relation F(X) = X for X a whole power of 2.Base Case: 8x2 P(X,0) = X Inductive Rule: 8x,y2, y>0, P(x,y) = P(x,y-1) + 1Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44For k = 0 to 3 P(k,0)=k For j = 1 to 7 For k = 0 to 3 P(k,j) = P(k,j-1) + 1Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Procedure P(x,y): If y=0 return x Otherwise return P(x,y-1)+1;Slide 56Slide 57Slide 58Giuseppe Peano [1889] Axiom’s For Slide 60Lemma: 0 + x = xLemma: Sx + y = S(x+y)Theorem: Commutative Property Of Addition: x + y = y + xSlide 64Aristotle’s ContrapositiveSlide 66Advice from the master.Contrapositive DominosContrapositive or Least Counter-Example Induction to Prove k, Sk“Strong” Induction To Prove k, SkAll Previous Induction To Prove k, SkSlide 72Slide 73Rene Descartes [1596-1650] “Method Of Infinite Decent”Slide 75Theorem: Each natural has a unique factorization into primes written in non-decreasing order.Slide 77Theorem: Each natural has a unique factorization into primes written in non-decreasing order.Slide 79Slide 80Slide 81“Strong” Induction Can Be Repackaged As Standard InductionSlide 83Slide 84Invariant Induction Suppose we have a time varying world state: W0, W1, W2, … Each state change is assumed to come from a list of permissible operations. We seek to prove that statement S is true of all future worlds.Invariant Induction Suppose we have a time varying world state: W0, W1, W2, … Each state change is assumed to come from a list of permissible operations.Odd/Even Handshaking Theorem: At any party at any point in time define a person’s parity as ODD/EVEN according to the number of hands they have shaken. Statement: The number of people of odd parity must be even.Slide 88Slide 89Inductive Definition of T(n)Slide 91Closed Form Definition of G(n)Two equivalent functions?Prove equivalence by induction on n: Assume T(x) = G(x) for x < nSlide 95Solving Recurrences Guess and VerifyStudy BeeInduction: One Step At A TimeGreat Theoretical Ideas In Computer ScienceSteven RudichCS 15-251 Spring 2004Lecture 4 Jan 22, 2004 Carnegie Mellon UniversityLast time we talked about different ways to represent numbers: unary, binary, decimal, base b, plus/minus binary, Egyptian binary . . .Different representations had different advantages and disadvantages.Today we will talk about INDUCTIONInduction is the primary way we:1.Prove theorems2.Construct and define objectsRepresenting a problem or object inductively is one of the most fundamental abstract representations.Let’s start with dominoesDomino Principle: Line up any number of dominos in a row; knock the first one over and they will all fall.n dominoes numbered 1 to nFk ´ The kth domino will fallIf we set them all up in a row then we know that each one is set up to knock over the next one:For all 1· k < n:Fk ) Fk+1n dominoes numbered 1 to nFk ´ The kth domino will fallFor all 1· k < n:Fk ) Fk+1F1 ) F2 ) F3 ) …F1 ) All Dominoes Falln dominoes numbered 0 to n-1Fk ´ The kth domino will fallFor all 0· k < n-1:Fk ) Fk+1F0 ) F1 ) F2 ) …F0 ) All Dominoes FallThe Natural Numbers = { 0, 1, 2, 3, . . .}Plato: The Domino Principle works for an infinite row of dominoesAristotle: Never seen an infinite number of anything, much less dominoes.Plato’s DominoesOne for each natural numberAn infinite row, 0, 1, 2, … of dominoes, one domino for each natural number. Knock the first domino over and they all will fall.Proof: Suppose they don’t all fall. Let k>0 be the lowest numbered domino that remains standing. Domino k-1¸0 did fall, but k-1 will knock over domino k. Thus, domino k must fall and remain standing. Contradiction.The Infinite Domino PrincipleFk ´ The kth domino will fallAssume we know thatfor every natural number k,Fk ) Fk+1F0 ) F1 ) F2 ) …F0 ) All Dominoes FallMathematical Induction: statements proved instead of dominoes fallenInfinite sequence of statements: S0, S1, …Fk ´ Sk provedInfinite sequence ofdominoes.Fk ´ domino k fellEstablish 1) F02) For all k, Fk ) Fk+1Conclude that Fk is true for all kInductive Proof / ReasoningTo Prove k, SkEstablish “Base Case”: S0Establish that k, Sk ) Sk+1Assume hypothetically that Sk for any particular k; Conclude that Sk+1k, Sk ) Sk+1Inductive Proof / ReasoningTo Prove k, SkEstablish “Base Case”: S0Establish that k, Sk ) Sk+1“Induction Hypothesis” SkUse I.H. to show Sk+1k, Sk ) Sk+1Inductive Proof / ReasoningTo Prove k¸b, SkEstablish “Base Case”: SbEstablish that k¸b, Sk ) Sk+1Assume k¸ bAssume “Inductive Hypothesis”: Sk Prove that Sk+1 followsWe already know that n, n= 1 + 2 + 3 + . . . + n-1 + n = n(n+1)/2. Let’s prove it by induction:Let Sn ´ “n =n(n+1)/2”Sn ´ “n =n(n+1)/2”Use induction to prove k¸0, SkEstablish “Base Case”: S0. 0=The sum of the first 0 numbers = 0. Setting n=0 the formula gives 0(0+1)/2 = 0. Establish that k¸0, Sk ) Sk+1“Inductive Hypothesis” Sk: k =k(k+1)/2 k+1 = k + (k+1) = k(k+1)/2 + (k+1) [Using I.H.] = (k+1)(k+2)/2 [which proves Sk+1]Induction is also how we can define and construct our world.So many things, from buildings to computers, are built up stage by stage, module by module, each depending on the previous stages.Well, almost alwaysInductive Definition Of Functions Stage 0, Initial Condition, or Base Case:Declare the value of the function on some subset of the domain. Inductive RulesDefine new values of the
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