# 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.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
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..
why?
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
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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