# 0.6 Discrete structures recursion  (Page 5/8)

 Page 5 / 8

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) .

#### Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10768/1.1
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 By Janet Forrester By Jemekia Weeden By OpenStax By Yasser Ibrahim By By Ann Schlosser By Christine Zeelie By OpenStax By Jonathan Long By Olivia D'Ambrogio