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Proof: Assume that for all positive integers k, n>k>1, k can be written as the product of prime numbers.

We are going to prove that n can be written as the product of prime numbers.

Since n is an integer, it is either a prime number or not a prime number. If n is a prime number, then it is the product of 1, which is a prime number, and itself. Therefore the statement holds true.

If n is not a prime number, then it is a product of two positive integers, say p and q. Since both p and q are smaller than n, by the induction hypothesis they can be written as the product of prime numbers (Note that this is not possible, or at least very hard, if the First Principle is being used). Hence n can also be written as the product of prime numbers.

Questions and exercises

1. Indicate which of the following statements are correct and which are not.

a. The number 23 can be generated for EI in Example 3 in Section Recursive Definition.

b. Basis and Inductive Clauses are sufficiency for membership for the set.

c. The set {4} can replace the basis for NE of Example 2 in Section Recursive Definition.

d. If empty set is the basis of S in Example 4 in Section Recursive Definition, then the string ab is in S.

2. Indicate which of the following statements are correct and which are not.

a. Algorithm 2 in Section Recursive Algorithm produces 0, 2, 6 and 8 when computing the third power of 2.

b. Recursive algorithms are good because they run more efficiently than iterative ones.

c. In Algorithm 3 in Section Recursive Algorithm, x is first compared with the key at the middle of L.

d. If the input to Algorithm 1 in Section Recursive Algorithm is not a natural number, then 0 is returned.

3. Look at the Section Mathematics Induction; indicate which of the following statements are correct and which are not.

a. In the Inductive Step, P(n) is proven assuming that P holds for the parent of n.

b. In the Inductive Step, since we assume P(k) for an arbitrary k, P(k+1) holds.

c. The Induction Hypothesis does NOT assume P(k) for all k.

d. In the Induction, since k is arbitrary, we can prove P(6) assuming P(5) holds.

e. The Basis Step proves the statement for the elements of the basis.

4. Look at the Section Mathematics Induction; indicate which of the following statements are correct and which are not.

a. In the Second Principle, P(k) is assumed true for one arbitrary value of k.

b. The Second Principle does not make a proof any easier.

c. The Basis Step of the First Principle is implicitly proven by the Second Principle.

d. The Second Principle can be applied when n starts at some integer larger than 0.

e. The Second Principle gives you more assumptions to use, making a proof easier.

5. Let A i = { 1, 2, 3, ..., i } for i = 1, 2, 3, ... . Find i = 1 n A i size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } {} A rSub { size 8{i} } } {}

6. Let A i = { i, i+1, i+2, ... } for i = 1, 2, 3, ... . Find intersect i = 1 n A i size 12{ intersect rSub { size 8{i=1} } rSup { size 8{n} } {A rSub { size 8{i} } } } {}

7. Give a recursive definition of the set of positive integers that are multiples of 5.

8. Give a recursive definition of

a. the set of even integers.

b. the set of positive integers congruent to 2 modulo 3.

c. the set of positive integers not divisible by 5.

9. When does a string belong to the set A of bit strings (i.e. strings of 0's and 1's) defined recursively by

Basis Clause: ∅ ∈ A

Inductive Clause: 0 x 1 ∈ A   if x A

where ∅ is the empty string (An empty string is a string with no symbols in it.)

Extremal Clause: Nothing is in A unless it is obtained from the Basis and Inductive Clauses.

10. Find   f(1) f(2) ,  and   f(3) ,   if   f(n) is defined recursively by   f(0) = 2   and   for n = 0, 1, 2, ...

a. f(n + 1) = f(n) + 2.

b. f(n + 1) = 3f(n).

c. f(n + 1) = 2 f(n) .

11. Find   f(2) ,   f(3) ,  and   f(4) ,   if   f(n) is defined recursively by   f(0) = 1 ,   f(1) = -2   and  for n= 1, 2,...

a. f(n + 1) = f(n) + 3f(n - 1).

b. f(n + 1) = f(n) 2 f(n - 1).

12. Let F be the function such that F(n) is the sum of the first n positive integers.  Give a recursive definition of F(n) .

13. Give a recursive algorithm for computing nx whenever n is a positive integer and x is an integer.

14. Give a recursive algorithm for finding the sum of the first n odd positive integers.

15. Use mathematical induction to prove that 3 + 3 * 5 + 3 * 52+ ... + 3 * 5n = 3 (5n+1 - 1)/4 whenever n is a nonnegative integer.

16. Prove that 12 + 32 + 52+ ... + (2 n + 1)2 = ( n + 1)(2 n + 1)(2 n + 3)/3 whenever n is a nonnegative integer.

17. Show that 2n> n 2whenever n is an integer greater than 4.

18. Show that any postage that is a positive integer number of cents greater than 7 cents can be formed using just 3-cent stamps and 5-cent stamps.

19. Use mathematical induction to show that 5 divides n 5- n whenever n is a nonnegative integer.

20. Use mathematical induction to prove that if A 1, A 2, ... A n are subsets of a universal set U , then i = 1 n A ¯ i size 12{ {overline { union rSub { size 8{i=1} } rSup { size 8{n} } {A} }} rSub { size 8{i} } } {}   =  intersect i = 1 n A ¯ i size 12{ intersect rSub { size 8{i=1} } rSup { size 8{n} } { {overline {A}} } rSub { size 8{i} } } {}

21. Find a formula for

1/2 + 1/4 + 1/8 + ... + 1/2n

by examining the values of this expression for small values of n . Use mathematical induction to prove your result.

22. Show that if   a 1, a 2, ..., a n   are n distinct real numbers, exactly n - 1 multiplications are used to compute the product of these n numbers no matter how parentheses are inserted into their product.  ( Hint : Use the second principle of mathematical induction and consider the last multiplication).

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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. Jan 23, 2008 Download for free at http://cnx.org/content/col10513/1.1
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