Mathematics for Biotechnology and Data Science
Chapter 3: Application of Set Theory
Chapter 6: MAPPING OR FUNCTIONS
Chapter 8: Inverse of a Function
Chapter 3: Application of Set Theory
- Database and Information Systems
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Field: Computer Science & Software Engineering
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Application: In database design (e.g., SQL), set theory is used to organize and manage data. Operations like union, intersection, and difference correspond to combining or filtering records in tables.
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Example: In an engineering inventory system, the set of available components can be intersected with the set of required components to identify which parts are ready for use in manufacturing.
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Field: Electronics & Communication Engineering
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Application: The design of logic circuits uses Boolean algebra, which is built on set theory principles. Logic gates such as AND, OR, and NOT correspond to set operations intersection, union, and complement.
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Example: In signal processing, combining two signals using an AND gate is equivalent to taking the intersection of two sets where both signals are active.
In electrical engineering, set theory is applied to represent and manipulate collections of components, signals, or states. For example, when analyzing digital circuits, logic gates operate on sets of binary inputs and outputs, and truth tables can be expressed in terms of set operations such as union, intersection, and complement. This helps in simplifying Boolean expressions and optimizing circuit layouts.
In biotechnology, set theory is used to manage and analyze large biological datasets, such as DNA or protein sequences. For instance, when comparing genetic sequences, sets can represent unique genes or protein markers, and operations like intersection help identify common genes across species, while differences reveal genetic variations. This is vital for applications such as disease marker identification, evolutionary studies, and synthetic biology design.
EXAMPLES 1: In a group of 800
people, 550 can speak Hindi and 450 can speak English. How many can speak both
Hindi and English?
SOLUTION:
Let H denote the set of people speaking Hindi and E denote the set of people
speaking English. We are given that
and 
We have to find 
.
We know that
Hence, 200 persons can speak both Hindi and English.
EXAMPLES 2: In
a group of 50 people, 35 speak Hindi, 25 speak both English and Hindi and all
the people speak at least one of the two languages. How many people speak only
English and not Hindi? How many people speak English?
SOLUTION: Let H denote the set of
people speaking Hindi and E the set of people speaking English. Then it is
given that
Now,
Thus, the number of people speaking English but not Hindi is 15.
We have
Hence the number of people who speak English 40.
EXAMPLES 3: In
a survey of 700 students in a college, 180 were listed as drinking Coffee, 275
as drinking Tea and 95 were listed as both drinking Coffee as well as Tea. Find
how many students were drinking neither Coffee nor Tea.
SOLUTION: Let
U be the set of all surveyed students, A denote the set if students drinking
Coffee and B be the students drinking Tea.
It is given that
We have to find 
Now
EXAMPLES 4: In
a class of 35 students, 17 have taken mathematics, 10 have taken mathematics
but not economics. Find the number of students who have taken both mathematics
and economics and the number of student who have taken economics but not
mathematics if it is given that each student has taken either mathematics or
economics or both.
SOLUTION:
Let A denote the set of students who have taken mathematics and B be the set of
students who have taken economics.
It is given that
Now, 
Thus 7 students have
taken both mathematics and economics.
Now, 
Now, 
Thus 18 students have taken economics but not mathematics.
EXAMPLES 5: In a town of 10,000 families it was found that
40% families buy newspaper A, 20% families buy newspaper B and 10% families buy
newspaper C. 5% families buy A and B, 3% buy B and C and 4% buy A and C.
If 2%
families buy all three news papers, find the number of families which
buy (i) A only (ii) B only (iii) none of A,B and C.
SOLUTION:
Let P,Q and R be the sets of families buying newspaper A,B and C respectively.
Let U be the universal
set Then,
of 10,000
of 10,000
of 10,000
of 10,000
of 10,000
of 10,000
of 10,000
(i)
Required number 
(ii)
Required number 
)
(iii)
Required number 
EXAMPLES 6: A college awarded 38 medals in Football, 15 in
Basketball and 20 to Cricket. If these medals went to a total of 58 men and
only three men got medals in all three sports, how many received medal in
exactly two of three sports?
SOLUTION: Let
F denote the set of men who received medals in Football, B the set of men who
received medals in Basketball and C the set of men who received medals on
Cricket then we have
Now,
Now Number of men who
received medals in exactly two of the three sports
Thus 9 men received
medals in exactly two of three sports.
EXERCISE