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    F<- 2 * 3

    i<- 3 + 1

producing F = 6 and i = 4.

Since i = 4, the while loop is not entered any longer, F = 6 is returned and the algorithm is terminated.

To prove that the algorithm is correct, let us first note that the algorithm stops after a finite number of steps. For i increases one by one from 1 and n is a positive integer. Thus i eventually becomes equal to n.

Next, to prove that it computes n!, we show that after going through the loop k times, F = k ! and i = k + 1 hold. This is a loop invariant and again we are going to use mathematical induction to prove it.

Proof by induction.

Basis Step: k = 1. When k = 1, that is when the loop is entered the first time, F = 1 * 1 = 1 and i = 1 + 1 = 2. Since 1! = 1, F = k! and i = k + 1 hold.

Induction Hypothesis: For an arbitrary value m of k, F = m! and i = m + 1 hold after going through the loop m times.

Inductive Step: When the loop is entered (m + 1)-st time, F = m! and i = (m+1) at the beginning of the loop. Inside the loop,

    F<- m!* (m + 1)

    i<- (m + 1) + 1

producing F = (m + 1)! and i = (m + 1) + 1.

Thus F = k! and i = k + 1 hold for any positive integer k.

Now, when the algorithm stops, i = n + 1. Hence the loop will have been entered n times. Thus F = n! is returned. Hence the algorithm is correct.

Mathematical induction -- second principle

There is another form of induction over the natural numbers based on the second principle of induction to prove assertions of the form ∀x P(x). This form of induction does not require the basis step, and in the inductive step P(n) is proved assuming P(k)   holds for all k<n . Certain problems can be proven more easily by using the second principle than the first principle because P(k) for all k<n can be used rather than just P(n - 1) to prove P(n).

Formally the second principle of induction states that

      if ∀n [ ∀k [ k<n size 12{ rightarrow } {} P(k) ] size 12{ rightarrow } {} P(n) ] , then ∀n P(n) can be concluded.

Here ∀k [ k<n size 12{ rightarrow } {} P(k) ] is the induction hypothesis.

The reason that this principle holds is going to be explained later after a few examples of proof. Example 1: Let us prove the following equality using the second principle:

For any natural number n , 1 + 3 + ... + ( 2n + 1 ) = ( n + 1 )2.

Proof: Assume that 1 + 3 + ... + ( 2k + 1 ) = ( k + 1 )2   holds for all k,   k<n.

Then 1 + 3 + ... + ( 2n + 1 ) = ( 1 + 3 + ... + ( 2n - 1 ) ) + ( 2n + 1 )

= n2 + ( 2n + 1 ) = ( n + 1 )2 by the induction hypothesis.

Hence by the second principle of induction 1 + 3 + ... + ( 2n + 1 ) = ( n + 1 )2   holds for all natural numbers.

Example 2: Prove that for all positive integer n, i = 1 n size 12{ Sum rSub { size 8{i=1} } rSup { size 8{n} } {} } {} i ( i! ) = ( n + 1 )! - 1

Proof: Assume that

1 * 1! + 2 * 2! + ... + k * k! = ( k + 1 )! - 1   for all k,   k<n.

Then 1 * 1! + 2 * 2! + ... + ( n - 1 ) * ( n - 1 )! + n * n!

= n! - 1 + n * n!    by the induction hypothesis.

= ( n + 1 )n! - 1

Hence by the second principle of induction i = 1 n size 12{ Sum rSub { size 8{i=1} } rSup { size 8{n} } {} } {} i ( i! ) = ( n + 1 )! - 1   holds for all positive integers.

Example 3: Prove that any positive integer n, n>1, can be written as the product of prime numbers.

Questions & Answers

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Maciej
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s. Reply
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s.
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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s.
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s. Reply
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s.
Graphene has a hexagonal structure
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Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Cesar
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Uday
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AMJAD
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Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
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AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jan 23, 2008 Download for free at http://cnx.org/content/col10513/1.1
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