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Working with three sets is similar as working with two sets. The underlying characteristics of set operations are same. The union and intersection of sets are carried out with three sets - one after another. Notably, union and intersection operations are commutative. This allows us to extend these set operations to third set in any sequence. Venn’s diagram enables us to visualize resulting set. In this module, we shall first formulate expression for the numbers in the union of three sets. Subsequently, we shall apply the formulation to real time analysis - using both graphical (Venn diagram) and analytical methods.

The union, involving three sets, can be considered in terms of union of a set with "union of other two sets". In that sense, union of three sets represent elements which belong to either of three sets. Here, we want to find the expression of numbers of elements in the union set, which is represented as :

n A B C

Before, we work on the expansion of this term, let us first find out what does the term " A B C " represent on Venn’s diagram? The figure below shows the representation of this term :

Union of three sets

The union set represent elements of three sets combined together.

The set shown in the figure above consists of following class of elements :

  • The elements, which are exclusive of sets “A”, “B” and “C” respectively.
  • The elements, which are common to a pair of two sets at a time.
  • The elements, which are common to all three sets.

In the figure below, the common areas between a pair of two sets are marked “1”, “2” and “3”. The common area among all three sets is marked “4”.

Union of three sets

The union consists of disjointed regions.

Union of three sets

As discussed earlier, the sum of numbers of individual sets is greater than the number of elements in “ A B C ”, unless the sets are disjoint sets. It is imperative that we account for the repetition of common elements. Proceeding as in the case of union of two sets, we deduct the intersections between each pair of sets as :

n A B C = n A + n B + n C - n A B - n A C - n B C

In this manner, we account for common elements between two sets. However, we have deducted elements "common to all three sets" in this process – three times. On the other hand, the elements "common to all three sets" are present in the numbers of each of the individual sets - in total three times as there are three sets. Ultimately, we find that we have not counted the elements common to all sets at all. It means that we need to account for the elements common to all three sets. In order to add this number, we first need to know – what does this common area (marked 4) represent symbolically?

In the earlier module, we have seen that the area marked “4” is represented by “ A B C ”. Hence, the correct expansion for the numbers of elements in the union, involving three set, is :

n A B C = n A + n B + n C - n A B - n A C - n B C + n A B C

This result is an important result as the same is used while studying probability.

Union of three sets (analytical method)

We can achieve this result analytically as well. Here, we consider “A” as one set and “ B C “ as other set. Then, we apply the relation, which has been obtained for the numbers in the union of two sets as :

Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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What is power set
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Period of sin^6 3x+ cos^6 3x
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Period of sin^6 3x+ cos^6 3x
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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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