Induction and Recursion. CMPS/MATH 2170: Discrete Mathematics
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1 Induction and Recursion CMPS/MATH 2170: Discrete Mathematics
2 Outline Mathematical induction (5.1) Sequences and Summations (2.4) Strong induction (5.2) Recursive definitions (5.3) Recurrence Relations (8.1)
3 Principle of Mathematical Induction Want to know if we can reach every step on a infinite ladder Suppose we know two things We can reach the first rung of the ladder If we can reach a particular rung of the ladder, then we can reach the next rung Can we conclude that we can reach every rung?
4 Mathematical Induction Want to show: " Z % : ' " Proof by induction on " Base case: verify that '(1) is true Inductive step: show that ' + ' for any + Z % Inductive hypothesis: Assume '(+) is true Want to prove ' is true
5 Examples Ex. 1: Prove that 3 divides! "! for any! Z & '!
6 Mathematical Induction Want to show: " Z % : ' " for " = /, / + 1, / + 2,, where / Z Proof by induction on " '(/) Base case: verify that '(1) is true Inductive step: show that ' + ' for any + Z % + Z 3"4 + / Inductive hypothesis: Assume '(+) is true Want to prove ' is true
7 Examples Ex. 1: Prove that 3 divides! "! for any! Z & Ex. 2: Prove! ' < 2 * for all integers! > 4 Ex. 3: Prove that a finite set with! elements has 2 * subsets Ex. 4: Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps.
8 More about Mathematical Induction Why Mathematical Induction is valid? Implied by the Weak Ordering Property: Every nonempty subset of Z " has a least element #(1) ( Z " : # ( # ( Z " : # - Pros Can be used to prove a wide variety of forall conjectures Easy to follow Cons Cannot be used to find new theorems Lack of insight
9 Outline Mathematical induction (5.1) Sequences and Summations (2.4) Strong induction (5.2) Recursive definitions (5.3) Recurrence Relations (8.1)
10 Sequences Informally, a sequence is an ordered list of objects List all positive even integers: 2, 4, 6, 8, 10, We can describe the sequence as! " " Z % where! " = 2( Formally, a sequence is a function with domain Z ) or N: the above sequene can be defined by +: Z ) Z ) ( 2(
11 Arithmetic Progression Consider a sequence: 1, 4, 7, 10, 13, 16, 19 We can represent it as! " = 1 +3 (, ( 0 An arithmetic progression is a sequence! " " N with! " =. +/ (, for., / R! 1 =.,! 2 =. + /,! 3 =. + 2/, 5: N R (. + /(
12 Geometric Progression Consider a sequence: 3, 6, 12, 24, 48, 96, We can represent it as! " = 3 2 ", ( 0 A geometric progression is a sequence! " " N with! " = -. ", for -,. R! 0 = -,! 1 = -.,! 2 = -. 2, 3: N R ( -. "
13 Summations For a sequence! ", we write & #! " =! * +!, + +! & "$* & #! " =! % +! %)* + +! & "$%
14 Summations Ex.1: Sums of Arithmetic Progressions * " N: & + = '() "(" + 1) 2 Ex.2: Sums of Geometric Progressions * " N: & '() 2 ' = 2* where 2 1 Proof by Mathematical Induction
15 Summations %& %& /! 2 " "#$ "#& "#& %& $ $! 2 " =! =! 2 "! 2 )*$ 2 " = 32! "#$ )#& )#& Index substitution: + =, 5 2 )
16 Outline Mathematical induction (5.1) Sequences and Summations (2.4) Strong induction (5.2) Recursive definitions (5.3) Recurrence Relations (8.1)
17 Strong Induction Want to prove: " Z % : ' " Proof by (weak) induction on ": Base case: verify that '(1) is true Inductive step: show that ' + ' for any + Z % Proof by strong induction on ": Base case: verify that '(1) is true Inductive step: show that [' 1 ' 2 ' + ] ' for any + Z %
18 Strong Induction A more general form of strong induction Want to prove:! " for " = $, $ + 1, $ + 2,, where $ Z Base step: verify that! $,! $ + 1,!($ + -) are true Inductive step: Assume [! $! $ + 1! 1 ] is true, prove! is true for every integer 1 $ + -
19 Examples of Strong Induction Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21,! " = 0,! & = 1,! ( =! ()& +! ()+,, 2 Initial conditions Recurrence relation This is called a recursive definition Fibonacci Tiling Ex.1:! ( 2 ( for all, 0 Ex. 2:! ( > 1 ()+ for any, 3 where 1 = (1 + 5)/ (golden ratio) Fibonacci Spiral
20 Outline Mathematical induction (5.1) Sequences and Summations (2.4) Strong induction (5.2) Recursive definitions (5.3) Recurrence Relations (8.1)
21 Recursively Defined Sequences Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21,! " = 0,! & = 1,! ( =! ()& +! ()+,, 2 A sequence of powers of 2: 1, 2, 4, 8, 16, 32 An explicit formula: 4 ( = 2 (,, 0 A recursive definition: 4 " = 2 ", 4 ( = 24 ()&,, 1 Initial conditions Recurrence relation Initial condition Recurrence relation
22 Recursively Defined Functions A recursive definition of!: N R, N = {0,1,2, 3 } Base step: specify! 0 Recursive step: specify! 5 in terms of! 0,! 1,,!(5 1), for any 5 1 Ex.1: Give a recursive definition of! 5 = 5! Ex.2: Give a recursive definition of! 5 = <, < < N is a given sequence
23 Recursively Defined Sets Consider a set! Z $ recursively defined by Base step: 3! Recursive step: if '! and (!, then ' + (!! is the set of all positive integers divided by 3
24 Outline Mathematical induction (5.1) Sequences and Summations (2.4) Strong induction (5.2) Recursive definitions (5.3) Recurrence Relations (8.1)
25 The Tower of Hanoi Task: Move the stack of disks from peg 1 to peg 3 subject to the following rules: Move one disk at a time Only the uppermost disk on a stack can be moved No disk can be placed on top of a smaller disk Question: how many steps are needed?
26 The Tower of Hanoi By André Karwath aka Aka - Own work, CC BY-SA 2.5,
27 The Tower of Hanoi! " - number of moves needed to solve the Tower of Hanoi with # disks # 1 disks! "/$ 1 move Divide & Conquer! $ = 1! " = 2! "/$ + 1, # 2 Initial condition Recurrence relation! "/$ How to find an explicit formula for! "? Expand the recurrence iteratively and then make a conjecture:! " = 2 " seconds 585 billion years Proof by mathematical induction
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