# Sets  (Page 2/2)

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$A=\left\{\text{x: x is a vowel in English alphabet}\right\}$

$B=\left\{\text{x: x is an integer and}\phantom{\rule{2pt}{0ex}}0

The roaster equivalents of two sets are :

$A=\left\{a,e,i,o,u\right\}$

$B=\left\{1,2,3,4,5,6,7,8,9\right\}$

Can we write set “B” as the one which comprises single digit natural number? Yes. Thus, we can see that there are indeed different ways to define and identify members and hence the flexibility in defining collection.

We should be careful in using words like “and” and “or” in writing qualification for the set. Consider the example here :

$C=\left\{\text{x:}x\in Z\phantom{\rule{2pt}{0ex}}\text{and}\phantom{\rule{2pt}{0ex}}2

Both conditional qualifications are used to determine the collection. The elements of the set as defined above are integers. Thus, the only member of the set is “3”.

Now, let us consider an example, which involves “or” in the qualification,

$C=\left\{\text{x:}\phantom{\rule{2pt}{0ex}}x\in A\phantom{\rule{2pt}{0ex}}\text{or}\phantom{\rule{2pt}{0ex}}x\in B\right\}$

The member of this set can be elements belonging to either of two sets "A" and "B". The set consists of elements (i) belonging exclusively to set "A", (ii) elements belonging exclusively to set "B" and (iii) elements common to sets "A" and "B".

## Example

Problem 1 : A set in roaster form is given as :

$A=\left\{\frac{{5}^{2}}{6},\frac{{6}^{2}}{7},\frac{{7}^{2}}{8}\right\}$

Write the set in “set builder form”.

Solution : We see here that we are dealing with natural numbers. The numerators are square of natural numbers in sequence. The number in denominator is one more than numerator for each member. We can denote natural number by “n”. Clearly, if numerator is “ ${n}^{2}$ ”, then denominator is “n+1”. Therefore, the expression that represent a member of the set is :

$x=\frac{{n}^{2}}{n+1}$

However, this set is not an infinite set. It has exactly three members. Therefore, we need to specify “n” so that only members of the set are exclusively denoted by the above expression. We see here that “n” is greater than 4, but “n” is less than 8. For representing three elements of the set,

$5\le n\le 7$

We can write the set, now, in the builder form as :

$A=\left\{x:\phantom{\rule{1em}{0ex}}x=\frac{{n}^{2}}{n+1},\text{where "n" is a natural number and}\phantom{\rule{1em}{0ex}}5\le n\le 7\right\}$

In set builder form, the sequence within the range is implied. It means that we start with the first valid natural number and proceed sequentially till the last valid natural number.

## Some important sets representing numbers

Few key number sets are used regularly in mathematical context. As we use these sets often, it is convenient to have predefined symbols :

• P(prime numbers)
• N (natural numbers)
• Z (integers)
• Q (rational numbers)
• R (real numbers)

We put a superscript “+”, if we want to specify membership of only positive numbers, where appropriate. " ${Z}^{+}$ ", for example, means set of positive integers.

## Empty set

An empty set has no member or object. It is denoted by symbol “φ” and is represented by a pair of braces without any member or object.

$\phi =\left\{\right\}$

The empty set is also called “null” or “void” set. For example, consider a definition : “the set of integer between 1 and 2”. There is no integer within this range. Hence, the set corresponding to this definition is an empty set. Consider another example :

$B=\left\{x:\phantom{\rule{1em}{0ex}}{x}^{2}=4\phantom{\rule{1em}{0ex}}\text{and x is odd}\right\}$

An odd integer squared can not be even. Hence, set “B” also does not have any element in it.

There is a bit of paradox here. If the definition does not yield an element, then the set is not well defined. We may be tempted to say that empty set is not a set in the first place. However, there is a practical reason to have an empty set. It enables mathematical operations. We shall find many examples as we study operations on sets.

## Equal sets

The members of two equal sets are exactly same. There is nothing more to it. However, we need to know two special aspects of this equality. We mentioned about repetition of elements in a set. We observed that repetition of elements does not change the set. Consider example here :

$A=\left\{1,5,5,8,7\right\}=\left\{1,5,8,7\right\}$

Another point is that sequence does not change the set. Therefore,

$A=\left\{1,5,8,7\right\}=\left\{5,7,8,1\right\}$

In the nutshell, when we have to compare two sets we look for distinct elements only. If they are same, then two sets in question are equal.

## Cardinality

Cardinality is the numbers of elements in a set. It is denoted by modulus of set like |A|.

Cardinality
The cardinality of a set “A” is equal to numbers of elements in the set.

The cardinality of an empty set is zero. The cardinality of a finite set is some positive integers. The cardinality of a number system like integers is infinity. Curiously, the cardinality of some infinite set can be compared. For example, the cardinality of natural numbers is less than that of integers. However, we can not make such deduction for the most case of infinite sets.

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
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x