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In real situation, we identify a collection with certain characteristic common to elements. For example, a set of students in a class is based on the characteristic that each student is member of that class. This type of interpretation, however, is generally restrictive and leads to misinterpretation. We tend to think that the collection is isolated in itself, which is obviously wrong.

We need to free our mind from thinking set as an isolated entity. Some of the students might be members of another collection like that of basketball team, whereas some others might be members of a particular house, say “Amity house” and so on

In the nutshell, we consider set as a collection, which has multiple intersections with other collections.

A set

A set has multiple intersections with other collections.


Problem 1: In the house of total 200 students, 140 students play basketball and 80 students play football. Each student of the house plays at least one of these two games. How many students play both basketball and football?

Solution : The individual sets here are students playing basket ball (B) and football (F). Hence,

n B = 140

n F = 80

Clearly, there is no bar that a students playing basketball can not play football. This is also evident from the sum of the numbers in each set. The sum is 140 + 80 = 220, whereas total numbers of students in the house is 200 only. Thus, there are students who play both games. We can interpret the total numbers as the union of two individual sets. Hence, applying expansion for the numbers of a union :

n A B = n A + n B n A B

The students who play both games constitute the intersection of two individual sets.

Putting values,

n B F = 140 + 80 200 = 20

Universal set and complement

Universal is inclusive of all related sets. If we observe the Venn’s diagram consisting of two individual sets, then we realize that largest closed region within the universal set is the union involving two sets i.e (A∪B). This union, however, is a subset of U. There is remaining area within the universal set, which is called the component of this union.

Now we know that a union represents elements which belong to either set exclusively or belong commonly with other sets. It means that the complement of union represents the region, which can not be defined by the characterizing criteria of the union. This complement of union, therefore, represents situations which is described in terms of “neither or nor” type. Actually, this set is given by De-morgan’s first law.


Problem 2: In a house of total 200 students, 100 students play basketball, 60 students play football and 20 play both games. How many students play neither basketball nor football?

Solution : We have already discussed that “neither nor” condition is same as that of De-morgan’s first law :

n B F = n B F

Now expanding the right hand term, we have :

n B F = n B F = U n B F

Further using formula for the numbers in a union,

n B F = U n B n F + n B F

Putting values,

n B F = 200 100 60 + 20 = 60

This is the required answer. However, there remains a question : why do we consider total numbers of students as the numbers in universal set, “U”, unlike previous example in which this number corresponds to numbers in the union of individual sets. Remember, earlier question had the phrase “Each student of the house plays at least one of these two games”. This ensured that total numbers represented the union as everyone was playing one of two games. Such restriction is not there in this example. In fact, we saw that there are students who are not playing either of two games at all! Thus, total number represents universal set in this example.


Union of two sets “A” and “B” conveys the meaning of consisting three categories of elements (i) elements exclusively belonging to “A” (ii) elements exclusively belonging to “B” and (iii) (i) elements commonly belonging to “A” and “B”. In totality, we see that union conveys the meaning of “or” – the elements may belong either to a particular set or to both sets.


Problem 3: In a group of students, 40 students study either English or Mathematics. Of these 25 students study Mathematics, 10 students study both Mathematics and English. How many students study English?

Solution : The word “or” in the first sentence indicates that union of students studying Mathematics (M) or English (E) or both is 40. Using formula, we have :

n M E = n M + n E n M E

n E = n M E n M + n M E

Putting values,

n E = 40 25 + 10 = 25


In the case of intersection of two sets, we have noted that difference represents the exclusive or isolated set, which is not common to other set. From the Venn’s diagram, we also observe that a given set is actually composed of two sets (i) difference set and (ii) intersection set.

n A = n A B + n A B


n B = n B A + n A B


Problem 4: In a house of 200 students, 120 students study Mathematics, 60 students study English and 40 students study both Mathematics and English. Find in the house : (i) students who study Mathematics but not English (ii) students who study English, but not Mathematics (iii) students who study either Mathematics or English and (iv) students who neither study Mathematics nor English.

Solution : Let us first characterize collections as given in the question. Two sets are given one for those who study Mathematics (M) and other for those who study English(E). The addition of numbers of individual sets is 120 + 60 = 180, which is less than total numbers of students. Hence, total numbers of 200 corresponds to universal set. Here,

U = 200 ; n M = 120 ; n E = 60 a n d n M E = 40.

(i) Students studying Mathematics, but not English means that we need to find the numbers in the difference of set i.e M – E.

n M E = n M n M E

n M E = 120 40 = 80

(ii) Students studying Mathematics, but not English means that we need to find the numbers in the difference of set i.e E – M.

n E M = n E n M E

n E M = 60 40 = 20

(iii) Students who study either Mathematics or English is equal to the numbers in the union of two sets.

n M E = n M + n E n M E

Putting values,

n M E = 120 + 60 40 = 140

(iv) Students who study neither Mathematics nor English is equal to the numbers in the compliment of the union of two sets.

n M E = U n M E = 200 140 = 60

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
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I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
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how to synthesize TiO2 nanoparticles by chemical methods
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how did you get the value of 2000N.What calculations are needed to arrive at it
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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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