<< Chapter < Page Chapter >> Page >

n A B C = n A + n B C n [ A B C ]

Applying result for the union of two sets for “ n B C ”, we have :

n B C = n B + n C n B C

Putting in the expression for “ n A B C ”,

n A B C = n A + n B + n C n B C n [ A B C ]

At this stage, our task is to evaluate “ n [ A B C ] ”, Recall that we have worked with the distributive property of “intersection operator over union operator”. Following distributive property,

n [ A B C ] = n [ A B A C ]

We can treat each of the terms in the small bracket on the right hand side of the above equation as a set. Applying relation obtained for the numbers in the union of two sets again, we have :

n [ A B C ] = n A B + n A C n [ A B A C ]

The last term in above equation is :

[ A B A C ] = A B C

Hence,

n [ A B C ] = n A B + n A C n [ A B C ]

Now, putting this expression in the expression of the numbers in the union involving three sets and rearranging terms, we have :

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

In the nutshell, we find that numbers of elements in the union, here, is equal to the sum of numbers in the individual sets, minus elements common to two sets taken at a time, plus elements common to all three sets.

Illustration

In this section, we shall work with an example, which is quite intuitive of the analysis, involving three sets. We shall see that analysis of set operations in terms of Venn’s diagram is very direct and simple. As such, we shall first attempt analyze situation with Venn’s diagram.

However, we need to emphasize that extension of set concepts to calculus, probability and other branches of mathematics require that we develop analytical skill with respect to set operations. Keeping this aspect in mind, we shall also work the solution, using analytical method.

Problem : In a town, a total 100000 people read newspaper. Out of these, 40 % read newspaper “A”, 30 % read newspaper “B”, 10 % read newspaper “C”. It is found that 5% read both “A” and “B”; 4% read both “A” and “C”; and 3% read both “B” and “C”. Also, 2% of the people read all three newspapers. Find numbers (i) who read only “A” (ii) who read only “B” (iii) who read neither of three newspapers.

We define three sets “A”, “B” and “C”, corresponding to people reading newspapers “A”, “B” and “C” respectively. From question, we have :

n A = 0.4 X 100000 = 40000

n B = 0.3 X 100000 = 30000

n C = 0.1 X 100000 = 10000

n A B = 0.05 X 100000 = 5000

n A C = 0.04 X 100000 = 4000

n B C = 0.03 X 100000 = 3000

n A B C = 0.2 X 100000 = 2000

Venn's diagram method

We observe that sum of the individual sets is less than total numbers of people , reading newspaper, in the town. Hence, total reading population represents universal set, “U”. The representations of these sets are shown on Venn’s diagram. Note that we have split the elements common to a pair of two sets in two parts (a) elements exclusive to intersection of two sets and (b) elements common to all three sets.

Union of three sets

The regions common to sets.

From the diagram,

(i) The required set is the region of “A” not common to “B” and “C”. This region represents elements, which are exclusive to set “A”. Thus, numbers of people reading only “A” is :

n 1 = 40000 3000 + 2000 + 2000 = 33000

(ii) The required set is the region of “B” not common to “A” and “C”. This region represents elements, which are exclusive to set “B”. Thus, numbers of people reading only “B” is :

n 2 = 30000 3000 + 1000 + 2000 = 24000

(iii) The required set is the remaining region of universal set “U” i.e. complement of the union of three sets. Now, proceeding as before, people who read newspaper “C” only is (see Venn’s diagram above):

n 3 = 100000 2000 + 1000 + 2000 = 5000

Hence, the required number is :

Union of three sets

The regions representing people, who read neither of three newspapers.

n 4 = 100000 33000 + 24000 + 5000 + 3000 + 2000 + 1000 + 2000

n 4 = 30000

Analytical method

(i) Here we are required to find the numbers of people reading only “A”. It is clear that this set is part of the people, who do not read newspapers “B” and “C”. As discussed in the case of two sets, the numbers of people who read neither “B” and “C” is given by De-morgan’s first equation,

B C = B C

Intersection of two complement sets

The region representing people, who read neither of two newspapers.

For clarity, we have shown this region in the Venn’s diagram. We should realize that this intersection of two sets also includes people who read newspaper “A”. However, we are required to know numbers of people, who read newspaper “A” only. The exclusion people reading other newspaper as well are not part of our required set.

The remaining set (refer Venn’s diagram) represent area consisting of people who exclusively read newspaper “A” or who do not read any of three newspapers. Now, we need the intersection of set A with the remaining region to obtain the numbers, who read only newspaper “A”. Hence, required number is :

n 1 = n A B C

Using De-morgan's equation,

n 1 = n A B C = n [ A B C ]

n 1 = n [ A [ U B C ] ] = n [ A U A B C ] ]

n 1 = n [ A A B C ] = n A n [ A B C ]

Using distributive property of intersection over union,

n 1 = n A n [ A B A C ]

Using formula of expansion of union of two sets,

n 1 = n A [ n A B + n A C n [ A B A C ] ]

n 1 = n A [ n A B + n A C n A B C ]

We see that values of each term on the right hand side are given. Putting these values,

n 1 = 40000 [ 5000 + 4000 2000 ] = 33000

(ii) Here we are required to find the numbers of people reading only “B”. Proceeding as before,

n 2 = n B [ n B A + n B C n B A C ]

n 2 = n B [ n A B + n B C n A B C ]

Putting values,

n 2 = 30000 [ 5000 + 3000 2000 ] = 24000

(iii) In order to find the numbers of people, who read neither of three newspapers, we first find union of three sets. The union represents people who read either of these newspapers – one, two or all three. Clearly, people, who do not read either of these papers constitute a complement of this union.

n 3 = A B C = U A B C

Using expansion for the numbers in the union of three sets,

n 3 = U [ n A + n B + n C n A B n A C n B C + n A B C ]

Putting values, we have :

n 3 = 100000 [ 40000 + 30000 + 10000 5000 4000 3000 + 2000 ] = 30000

Questions & Answers

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.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
what is Nano technology ?
Bob Reply
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?
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 did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
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

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Functions' conversation and receive update notifications?

Ask