# 1.1 Subsets

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The collections are generally linked in a given context. If we think of ourselves, then we belong to a certain society, which in turn belongs to a province, which in turn belongs to a country and so on. In the context of a school, all students of a school belong to school. Some of them belong to a certain class. If there are sections within a class, then some of these belong to a section.

We need to have a mathematical relationship between different collections of similar types. In set theory, we denote this relationship with the concept of “subset”.

Subset
A set, “A” is a subset of set “B”, if each member of set “A” is also a member of set “B”.

We use symbol “ $\subset$ ” to denote this relationship between a “subset” and a “set”. Hence,

$A\subset B$

We read this symbolic representation as : set “A” is a subset of set “B”. We express the intent of relationship as :

$A\subset B\phantom{\rule{1em}{0ex}}\mathrm{if}\phantom{\rule{1em}{0ex}}x\in A,\phantom{\rule{1em}{0ex}}\mathrm{then}\phantom{\rule{1em}{0ex}}x\in B$

It is evident that set "B" is larger of the two sets. This is sometimes emphasized by calling set "B" as the “superset” of "A". We use the symbol “ $\supset$ ” to denote this relation :

$B\supset A$

If set "A" is not a subset of "B", then we write this symbolically as :

$A\not\subset B$

## Important results / deductions

Some of the important characteristics and related deductions are presented here :

## Equality of two sets

It is clear that set “B” is inclusive of subset “A”. It means that “B” may have additional elements over and above those common with “B”. In case, all elements of “B” are also in “A”, then two sets are equal. We express this symbolically as :

$If\phantom{\rule{1em}{0ex}}A\subset B\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}B\subset A,\phantom{\rule{1em}{0ex}}\mathrm{then}\phantom{\rule{1em}{0ex}}A=B.$

This is true in other direction as well :

$If\phantom{\rule{1em}{0ex}}A=B,\phantom{\rule{1em}{0ex}}\mathrm{then}\phantom{\rule{1em}{0ex}}A\subset B\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}B\subset A.$

We can write two instances in a single representation as :

$A\subset B\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}B\subset A⇔A=B$

The symbol “ $⇔$ ” means that relation holds in either direction.

## Relation with itself

Every set is subset of itself. This is so because every element is present in itself.

## Relation with empty set

Empty set is a subset of every set. This deduction is direct consequence of the fact that empty set has no element. As such, this set is subset of all sets.

## Proper subset

We have seen from the deductions above that special circumstance of “equality” can blur the distinction between “set” and “subset”. In order to emphasize, mother-child relation between sets, we coin the term “proper subset”. If every element of set “B” is not present in set “A”, then “A” is a “proper” subset of set “B”; otherwise not. This means that set “B” is a larger set, which, besides other elements, also includes all elements of set “A”.

Set of vowels in English alphabet, “V”, is a “proper” subset of set of English alphabet, “E”. All elements of “V” are present in “E”, but not all elements of “E” are present in “V”.

There is a bit of conventional differences. Some write a “proper” subset relation using symbol “ $\subset$ ” and write symbol “ $\subseteq$ ” to mean possibility of equality as well. We have chosen not to differentiate two subset types.

## Number system

The number system is one such system, in which different number groups are related. Natural number is a subset of integers. integers are subset of rational numbers and rational numbers are subset of real numbers. None of these sets are equal. Hence, relations are described by proper subsets.

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
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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
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Alexandre
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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
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LITNING
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what is differents between GO and RGO?
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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?
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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
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x By OpenStax By By Christine Zeelie By OpenStax By Prateek Ashtikar By Katy Pratt By OpenStax By Edgar Delgado By Marion Cabalfin By David Corey