Indksi Mat 2

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MATHEMATICAL INDUCTION



An illustration

Penjumlahan n buah bilangan ganjil pertama

1 = 1 (2.1-1) = 12

1 + 3 = 4 1 + (2.2-1) = 22

1 + 3 + 5 = 9 1 + 3 + (2.3-1) = 32

1 + 3 + 5 + 7 = 16 1 + 3 + 5 + (2.4-1) = 42

……………. ………………

……………. ………………

Hypothesis: 1 + 3 + 5 + 7 + 9 + ... + (2n-1) = n2 ??



A proof by Mathematical Induction

P(n) is true for every positive integer n:

1. Basic step.P(1) is shown to be true.

2. Induction step.

P(k) P(k+1) is shown to be true

for every positive integer k.



Hypothesis: 1 + 3 + 5 + 7 + 9 + ... + (2n-1) = n2 ??

Proof:

Basic Step. (2.1-1) = 12

1=1.1= 12 1 = 12 (2-1) = 12 (2.1-1) = 12

Induction step. 1 + 3 + 5 + 7 + 9 + ... + (2k-1) = k2 .

↓ 1 + 3 + 5 + 7 + 9 + ... + (2(k+1) -1) = (k+1) 2

Misalkan 1 + 3 + 5 + 7 + 9 + ... + (2k-1) = k2, maka:

1 + 3 + 5 + 7 + 9 + …………. + (2(k+1) -1)

= 1 + 3 + 5 + 7 + 9 + ... + (2k-1) + (2(k+1) -1)

= k2 + (2(k+1) -1)

= k2 + 2k+2 -1

= k2 + 2k+1

= (k+1) 2



Examples, show that :

1 + 21 + 22 + … + 2n = 2n+1 – 1 for all

nonnegative integers n.

n < 2n for all positive integers n.

Every amount of postage of 12 cents or

more can be formed using just 4 cent and5 cent stamps.



The second principle

of mathematical induction

1. Basic step. P(1) is shown to be true.

2. Inductive step. It is shown that

[P(1) and P(2) and … and P(k)] P(k+1)

is true for every positive integer k.



RECURSIVE DEFINITIONS

Recursively defined functions

To define a function with the set of

nonnegative integers as its domain,

1. Specify the value of the function at zero.

2. Give a rule for finding its value as an

integer from its values at smaller integer.



Examples:

1 Suppose f(0) = 3 and f(n+1) = 2.f(n)+3.

Find f(1), f(2), f(3), and f(4).

2 If f(0)=0, f(1)=1 and f(n)=f(n-1)+f(n-2).Then f(6)?

3 Give a recursive definition of an where a is

a nonzero real number and n is anonnegative integer.



Recursively Defined Sets

1.Let S be defined by:

3 S

if x

S and y

S, then x + y

S

2.Give a recursive definition of l(w), the

length of the string w.(Note that denotes for empty string)



Recursive Algorithms

Definition. An algorithm is called

recursive if it solves a problem by reducing

it to an instance of the same problem with

smaller input.



Examples:

1.Algorithm for computing an where a is a

nonzero real number and n is a

nonnegative integer.

Procedure power(a:nonzero real number,

n:nonnegative integer)

if n=0 then power(a,n):=1else power(a,n):=a*power(a,n-1)



2.Linier search algorithm as a recursive

procedure

Procedure search(i,j,x)if ai=x then location:=i

else if i = j then location:=0

else search(i+1,j,x)



Recursion and Iteration

A Recursive Procedure for Factorials

Procedure recursive factorial(n: positive integer)

if n=1 then factorial(n):= 1

else factorial(n):= n*factorial(n-1)

An Iterative Procedure for Factorial

Procedure iterative factorial(n: positive integer)x:= 1

for i:= 1 to n x:= i*x

{x is n!}

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