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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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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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