# 0.6 Discrete structures recursion  (Page 5/8)

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else

if ( w = w1 ∨w2 or w1 ⋀w2 or w1 $\to$ w2 or w1 $↔$ w2 ) and

Proposition(w1) = Yes and   Proposition(w2) = Yes

then return Yes

else return No

end

## Mathematical induction -- first principle

As we have seen in recursion, the set of natural numbers can be defined recursively, and its elements can be generated one by one starting with 0 by adding 1. Thus the set of natural numbers can be described completely by specifying the basis element (0), and the process of generating an element from a known element in the set.

Taking advantage of this, natural numbers can be proven to have certain properties as follows:

First it is proven that the basis element, that is 0, has the property in question (basis step). You prove that the seeds (the first generation elements) have the property. Then it is proven that if an arbitrary natural number, denote it by n, has the property in question, then the next element, that is n + 1, has that property (inductive step). Here you prove that the property is inherited from one generation (n) to the next generation (n + 1).

When these two are proven, then it follows that all the natural numbers have that property. For since 0 has the property by the basis step, the element next to it, which is 1, has the same property by the inductive step. Then since 1 has the property, the element next to it, which is 2, has the same property again by the inductive step. Proceeding likewise, any natural number can be shown to have the property. This process is somewhat analogous to the knocking over a row of dominos with knocking over the first domino corresponding to the basis step.

More generally mathematical statements involving a natural number n such as 1 + 2 + ... + n = n( n + 1 )/2 can be proven by mathematical induction by the same token.

To prove that a statement P(n) is true for all natural number n≥n0, where n0 is a natural number, we proceed as follows:

Basis Step: Prove that P(n0) is true.

Induction: Prove that for any integer k≥n0, if P(k) is true (called induction hypothesis), then P(k+1) is true.

The first principle of mathematical induction states that if the basis step and the inductive step are proven, then P(n) is true for all natural number n≥n0.

As a first step for proof by induction,   it is often a good idea to restate P(k+1) in terms of P(k) so that P(k), which is assumed to be true, can be used.

Example:

Prove that for any natural number n,   0 + 1 + ... + n = n( n + 1 )/2 .

Proof:

Basis Step: If n = 0, then LHS = 0, and RHS = 0 * (0 + 1) = 0 .

Hence LHS = RHS.

Induction: Assume that for an arbitrary natural number n, 0 + 1 + ... + n = n( n + 1 )/2 .

-------- Induction Hypothesis

To prove this for n+1,   first try to express LHS for n+1   in terms of LHS for n,   and somehow use the induction hypothesis.

Here let us try

LHS for n + 1 = 0 + 1 + ... + n + (n + 1) = (0 + 1 + ... + n) + (n + 1).

Using the induction hypothesis, the last expression can be rewritten as

n( n + 1 )/2 + (n + 1) .

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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Maciej
Abigail
for teaching engĺish at school how nano technology help us
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what is fullerene does it is used to make bukky balls
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what is the actual application of fullerenes nowadays?
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Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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