15 251 Some Great Theoretical Ideas in Computer Science for Stuffs Homework due Tonight Quiz on Thursday Inductive Reasoning Lecture 3 January 20 2009 Dominoes Domino Principle Line up any number of dominos in a row knock the first one over and they will all fall Plato The Domino Principle works for an infinite row of dominoes Aristotle Never seen an infinite number of anything much less dominoes Plato s Dominoes One for each natural number Theorem An infinite row of dominoes one domino for each natural number Knock over the first domino 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 Mathematical Induction statements proved instead of dominoes fallen Infinite sequence of dominoes Fk domino k fell Infinite sequence of statements S0 S1 Fk Sk proved Establish 1 F0 2 For all k Fk Fk 1 Conclude that Fk is true for all k Inductive Proofs To Prove k Sk Establish Base Case S0 Establish that k Sk Sk 1 k Sk Sk 1 Assume hypothetically that Sk for any particular k Conclude that Sk 1 Theorem The sum of the first n odd numbers is n2 Check on small values 1 1 3 1 3 5 1 3 5 7 1 4 9 16 Theorem The sum of the first n odd numbers is n2 The kth odd number is 1 when k 0 2k Sn is the statement that 1 3 5 2k 1 2n 1 n2 Establishing that n 1 S S n 1 3 5 2k 1 2n 1 n 2 n Base Case S1 Domino Property Assume Induction Hypothesis Sk That means 1 3 5 2k 1 1 3 5 2k 1 2k 1 k2 k2 2k 1 Sum of first k 1 odd numbers k 1 2 Theorem The sum of the first n odd numbers is n2 Primes A natural number n 1 is a prime if it has no divisors besides 1 and itself Note 1 is not considered prime Theorem Every natural number 1 can be factored into primes Sn n can be factored into primes Base case 2 is prime S2 is true How do we use the fact Sk 1 k 1 can be factored into primes to prove that Sk k can be factored into primes This shows a technical point about mathematical induction Theorem Every natural number 1 can be factored into primes A different approach Assume 2 3 k 1 all can be factored into primes Then show that k can be factored into primes All Previous Induction To Prove k Sk Establish Base Case S0 Establish Domino Effect Assume j k Sj use that to derive Sk All Previous Induction Also called To Prove k Sk Strong Induction Establish Base Case S0 Establish Domino Effect Assume j k Sj use that to derive Sk And there are more ways to do inductive proofs Method of Infinite Descent Rene Descartes Show that for any counterexample you find a smaller one If a counter example exists there would be an infinite sequence of smaller and smaller counter examples Theorem Every natural number 1 can be factored into primes Let n be a counter example Hence n is not prime so n ab If both a and b had prime factorizations then n would too Thus a or b is a smaller counter example Yet another way of packaging inductive reasoning is to define invariants Invariant n 1 Not varying constant 2 Mathematics Unaffected by a designated operation as a transformation of coordinates Invariant n 3 Programming A rule such as the ordering of an ordered list that applies throughout the life of a data structure or procedure Each change to the data structure maintains the correctness of the invariant 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 We seek to prove that statement S is true of all future worlds Argue that S is true of the initial world Show that if S is true of some world then S remains true after one permissible operation is performed 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 Statement The number of people of odd parity must be even Initial case Zero hands have been shaken at the start of a party so zero people have odd parity Invariant Argument If 2 people of the same parity shake they both change and hence the odd parity count changes by 2 and remains even If 2 people of different parities shake then they both swap parities and the odd parity count is unchanged Inductive reasoning is the high level idea Standard Induction All Previous Induction Least Counter example Invariants all just different packaging Induction Problem A circular track that is one mile long There are n 0 gas stations scattered throughout the track The combined amount of gas in all gas stations allows a car to travel exactly one mile The car has a very large tank of gas that starts out empty Show that no matter how the gas stations are placed there is a starting point for the car such that it can go around the track once 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 Inductive Definition Example Initial Condition or Base Case F 0 1 Inductive definition of the powers of 2 Inductive Rule For n 0 F n F n 1 F n 1 n 0 1 2 3 F n 1 2 4 8 4 5 6 7 16 32 64 128 Leonardo Fibonacci In 1202 Fibonacci proposed a problem about the growth of rabbit populations Rabbit Reproduction A rabbit lives forever The population starts as single newborn pair Every month each productive pair begets a new pair which will become productive after 2 months old Fn of rabbit pairs at the beginning of the nth month month 1 2 3 4 5 6 7 rabbits 1 1 2 3 5 8 13 Fibonacci Numbers month rabbit s 1 2 3 4 5 6 7 1 1 2 3 5 8 13 Stage 0 Initial Condition or Base Case Fib 1 1 Fib 2 1 Inductive Rule For n 3 Fib n Fib n 1 Fib n 2 Example T 1 1 T n 4T n 2 n Notice that T n is inductively defined only for positive powers of 2 and undefined on other values T 1 1 T 2 6 T 4 28 T 8 120 Guess a closed form formula for T n Guess G n 2n2 n Inductive Proof of Equivalence Base Case G 1 1 and …
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