Mathematics for Biotechnology and Data Science
Chapter 3: Application of Set Theory
Chapter 6: MAPPING OR FUNCTIONS
Chapter 8: Inverse of a Function
1] SETS AND THEIR REPRESENTATION
Definition of a Set: A well-defined collection of distinct objects is called Set.
Example 1: The collection of vowels in English alphabets. This set containing five elements, namely a,e,i,o,u.
Example 2: A collection of favourite singers is not a set because it varies from person to person.
If a is an element of a
set A, then we write a 
A and say a belongs to A or a is in A or a is
member of A.
If a does not belongs to A, we write a 
A.
Example 3: ‘a’ belongs to the set of vowels in English alphabets.
Example 4: The collection of all states in the Indian union is a set.
some standard well known set are used in study. so we reserve some letters for these sets as listed below:
:
for the set of natural numbers.
:
for the set of integers.
: for the set of
all positive integers
:
for the set of all rational numbers
:
for the set of all positive rational numbers
: for
the set of all real numbers
:
for the set of all positive real numbers.
:
for the set of all complex numbers.
Representation of Set : There are two ways to represent a set:
1) Roaster Method: In this method a set is described by listing elements, separated by commas, within { }.
Example 5: The
set of even natural numbers can be described as 
. Here the dots stand for ‘and so on’.
2)
Set-Builder
Method: In this method, a set is described by a characterizing property P(x) of
its elements x. In such a case the set is described by 
or 
, which is read as ‘ the set of all x such that P(x)
holds’, The symbol ‘|’ or ‘:’ is read as ‘such that’.
Example 5: The
set A 
can be written
as A 
.
2. Types of Sets
Empty set: A set is said to be empty or null or void set if it
has no elements and it is denoted by 
.
In Roaster method, is denoted as {
}. It follows from this definition that a set A is an empty set if the
statement x A is not true for any x.
Example 6: 
Example 7: 
A set consisting of at least one element is called a non-empty or non-void set.
Singleton set: A set consisting of a single element is called a singleton set.
Example 8: The set 
is a singleton
set.
Example 9: The set of even prime number is a singleton set.
Finite set: A set is called a finite set if it is either void set or its elements can be listed (counted, labelled) by natural numbers 1,2,3,… and the process of listing terminates at a certain natural number n (say).
Cardinal number of a finite set: The number n in
the above definition is called the cardinal number or order of a finite set A
and is denoted by 
Infinite set: A set whose elements cannot be listed by the natural numbers 1,2,3,…,n, for any natural number n is called an infinite set.
The infinite sets are denoted by using dots (. . .) at the end of the elements which follow a pattern. Any set A which is an infinite set can be represented as follows:
.
Example 10: Each one of the following is an example of finite set:
(1) Set of natural numbers less than 100.
(2) Set of soldiers in Indian Army.
(3) Set of even prime Natural number less than 2.
(4) Set of all persons on the earth.
Example 11: Each one of the following is an example of infinite set:
(1) Set of all real numbers on a number line.
(2) Set of all lines passing through a point.
Equivalent Sets: Two sets A and B are equivalent if their cardinal
numbers are same i.e., 
Equal Sets: Two sets A and B are said to be equal if every element ofA is a member of B, and every element of B is a member of A.
If
sets A and B are equal, we write A 
B and A 
B when A and B
are not equal.
Examples with solutions:
1) Example 12: Which of the following sets are empty sets?
i. A 
ii. B 
iii.C 
iv. D 
Solution: (i) We know that there is
no rational number whose square is 3. So, 
is not satisfied by any rational number.
Hence, A is an empty set.
(ii) We know that 2 is the only even prime number.
Therefore, B 
. So, B is not an empty
set.
(iii) Since there is no natural number between 4 and 5. So, C is an empty set.
(iv) Since 
satisfy 
and 
are odd integers. Therefore, D 
. Thus, D is a
non-empty set.
2) Example 13: Find the pairs of equal sets, from the following sets, if any, giving reasons:
A
,
B 
,
C 
,
D 
,
E
.
Solution: We have,
A 
,
B 
,
C 
,
D 
,
and E 
.
Clearly, C E.
3) Example 14: Which of the following pairs of sets are equal? Justify your answer.
i. A 
B 
ii. A
, B 
Solution: (i) We have, A 
and B 
in set order
doesn’t matter therefore A B.
(ii) We have, A 
and
B
Since
A but 
B. So, A B.
Example 15: State which of the following sets are finite and which are infinite:
i.
A 
ii.
B 
iii.
C 
iv.
D 
Solution: (i) we have, A 
. So, A is a finite
set.
(ii) We have,
B 
Clearly, B is an infinite set.
(iii) We
have, C 
Clearly, C is a finite set.
(iv) We have,
D 
Clearly, D is an infinite set.
Exercise 1
1) Which of the following are examples of empty set?
i. Set of all even natural numbers divisible by 5.
ii. Set of all even prime numbers.
iii.
iv.
Ans: (iii), (iv)
2) Which of the following sets are finite and infinite?
i. Set of concentric circles in a plane;
ii. Set of letters of the English Alphabets;
iii.
;
iv.
.
Ans: (i) Infinite, (ii) Finite, (iii) Infinite and (iv) Finite.
3) Which of the following sets are equal?
i.
A 
ii.
B 
iii.
C 
iv.
D 
Ans: A = C = D
4) Are the following sets are equal?
A
;
B
;
C
.
Ans: No
5) From the sets given below, pair the equivalent sets:
A
, B 
, C 
, D 
.
Ans: A,C ; B,D
6) From the sets given below, select equal sets and equivalent sets.
A
, B 
, C 
, D 
,
E
, F 
, G 
, H 
.
Ans: Equal sets: B = D, C = F ; Equivalent sets: A,E,H; B,D,G; C,F
2. SUBSET
Definition: Let A and B be two sets. If every element of A is an element of B, then A is called a subset of B.
If A is a subset of B, we write A 
B, which is read as “ A is a subset of B” or “
A is contained in B”, or B is superset of A.
Thus, A B if a A 
a B.
Obviously, every set is a subset of itself and the empty set is subset of every set.
These
two sets are called improper subsets. A subset A of a set B is called a proper subset of B if A B. and we write A 
B.
Note:
The total number of subsets of a finite set
containing n elements is 
Universal set
A set that contains all sets in a given context is called the universal set.
Power set
Let A be a set. Then the collection or family of all subsets of A is called the power set of A and is denoted by P(A).
That
is, P(A) 
.
Examples with solutions:
Example 16: Consider the following sets:
, A 
, B 
, C 
Insert
the correct symbol or 
between each of the following pair of sets:
(i)
…… B (ii) A……B (iii) A……C (iv) B……C
Solution:
i.
Since null set is subset of every set.
Therefore, B.
ii.
Clearly, 2 A but 2 B. So, A B.
iii.
Since all elements of set A are in C and
A C. So, A C.
iv.
Clearly, all elements of set B are in set
C and B C. So, B C.
Example 17: Let
A 
, B 
and C 
. Find all sets X such
that:
(i)
X B and X C
(ii)
X A and X B.
Solution: (i) We have,
P(A) 
P(B) 
and, P(C) 
Now, X B and X C
X P(B) and X P(C).
X 
(iii)
We have, X A and X B
X is a subset of A but X is not a subset of B.
X P(A) but X P(B)
X 
Example 18: Let
A, B and C be three sets. If A B and B C, is it true that A C? If not give an example.
Solution: Consider the following sets:
A 
, B 
and C 
.
Clearly, A B and B C. But, A C as a A but a C.
Thus, the given statement is not true.
Example 19: Let
A , B and C 
. Find all sets X
satisfying each pairs of condition:
i.
X B and X C
ii.
X B, X B and X C.
Solution:
(i) We have,
X B and X C
X is a subset of B but X is not a subset of C
X P(B) but x P(C)
X 
(ii) We have,
X B, X B and X C
X is a subset of B other than B itself and X
is not a subset of C
X P(B), x
P(C) and X B
X 
.
Example 20: Let
B be a subset of a set A and let P(A : B) 
i.
Show that P(A : ) =P(A)
ii.
If A 
and B . List all the members
of the set P(A : B).
Solution:
i. We have,
P(A : B) 
Set of all those subsets of A which contain B.
P(A : ) Set of all those subsets of A which contain B.
Set of all subsets of set A
P(A).
ii.
If A and B . Then,
P(A : B) Set of all those subsets of set A which
contain B
Example 21: Prove
that A 
implies A 
.
Solution:
We know that two sets A and B are equal iff A B and B A. Also,
We know that
A
And, A 
A .
Exercise 2
1) Which of the following statements are true? Give reason to support your answer.
i.
For any two sets A and B either A B or B A;
ii. Every subset of an infinite set is infinite;
iii. Every subset of a finite set is finite;
iv. Every set has a proper subset
v. A set can have infinitely many subsets.
Ans: (iii),(iv)
2) State whether the following statements are true or false:
i.
1 
ii.
a 
iii.
iv.
Ans: i. T ii. F iii. F iv. T
3) Decide among the following sets, which are subsets of which:
A
B
C
D
.
Ans:
DABC
4) Write down all possibles proper subsets each of the following sets:
i. 
, ii. 
, iii. 
5) What is the total number of proper subsets of a set containing of n elements?
Ans. 
6) If
A is any set, prove that A 
7) Prove
that : A 
B, B C and C A A C.
................ Continued .............
