SETS, RELATIONS & FUNCTIONS
SETS, RELATIONS & FUNCTIONS
SETS
1. SET 2.2 Set-Builder Form
A set is a collection of well-defined and well distinguished In this form, we write a variable (say x) representing any
objects of our perception or thought. member of the set followed by a property satisfied by each
member of the set.
1.1 Notations
For example, the set A of all prime numbers less than 10 in
The sets are usually denoted by capital letters A, B, C, etc. the set-builder form is written as
and the members or elements of the set are denoted by lower- A = {x | x is a prime number less that 10}
case letters a, b, c, etc. If x is a member of the set A, we write The symbol '|' stands for the words 'such that'. Sometimes,
x A (read as 'x belongs to A') and if x is not a member of the we use the symbol ':' in place of the symbol '|'.
set A, we write x A (read as 'x does not belong to A,). If x
and y both belong to A, we write x, y A. 3. TYPES OF SETS
2. REPRESENTATION OF A SET
3.1 Empty Set or Null Set
Usually, sets are represented in the following two ways :
A set which has no element is called the null set or empty
(i) Roster form or Tabular form
set. It is denoted by the symbol I .
(ii) Set Builder form or Rule Method
2.1 Roster Form For example, each of the following is a null set :
(a) The set of all real numbers whose square is –1.
In this form, we list all the member of the set within braces
(curly brackets) and separate these by commas. For example, (b) The set of all rational numbers whose square is 2.
the set A of all odd natural numbers less that 10 in the Roster (c) The set of all those integers that are both even and odd.
form is written as :
A set consisting of atleast one element is called a
A = {1, 3, 5, 7, 9}
non-empty set.
3.2 Singleton Set
A set having only one element is called singleton set.
For example, {0} is a singleton set, whose only member is 0.
(i) In roster form, every element of the set is listed
only once. 3.3 Finite and Infinite Set
(ii) The order in which the elements are listed is
A set which has finite number of elements is called a finite
immaterial.
set. Otherwise, it is called an infinite set.
For example, each of the following sets denotes
For example, the set of all days in a week is a finite set
the same set {1, 2, 3}, {3, 2, 1}, {1, 3, 2}
whereas the set of all integers, denoted by
{............ -2, -1, 0, 1, 2,...} or {x | x is an integer}, is an infinite set.
An empty set I which has no element in a finite set A is
called empty of void or null set.
, SETS, RELATIONS & FUNCTIONS
3.4 Cardinal Number 4. OPERATIONS ON SETS
The number of elements in finite set is represented by n(A), 4.1 Union of Two Sets
known as Cardinal number.
3.5 Equal Sets The union of two sets A and B, written as A B (read as 'A
union B'), is the set consisting of all the elements which are
Two sets A and B are said to be equals, written as A = B, if
every element of A is in B and every element of B is in A. either in A or in B or in both Thus,
3.6 Equivalent Sets A B = {x : x A or x B}
Clearly, x A B x A or x B, and
Two finite sets A and B are said to be equivalent, if n
(A) = n (B). Clearly, equal sets are equivalent but equivalent x A B x A and x B.
sets need not be equal.
For example, the sets A = { 4, 5, 3, 2} and B = {1, 6, 8, 9} are
equivalent but are not equal.
3.7 Subset
Let A and B be two sets. If every elements of A is an element
of B, then A is called a subset of B and we write A B or
B A (read as 'A is contained in B' or B contains A'). B is
called superset of A.
For example, if A = {a, b, c, d} and B = {c, d, e, f}, then
A B = {a, b, c, d, e, f}
4.2 Intersection of Two sets
(i) Every set is a subset and a superset itself.
The intersection of two sets A and B, written as A B
(ii) If A is not a subset of B, we write A B.
(read as ‘A’ intersection ‘B’) is the set consisting of all the
(iii) The empty set is the subset of every set. common elements of A and B. Thus,
(iv) If A is a set with n(A) = m, then the number of A B = {x : x A and x B}
subsets of A are 2m and the number of proper
subsets of A are 2m -1. Clearly, x A B x A and x B, and
For example, let A = {3, 4}, then the subsets of A x A B x A or x B.
are I , {3}, {4}. {3, 4}. Here, n(A) = 2 and number
of subsets of A = 22 = 4. Also, {3} {3,4}and {2,3}
{3, 4}
3.8 Power Set
The set of all subsets of a given set A is called the power set
of A and is denoted by P(A).
For example, if A = {1, 2, 3}, then
P(A) = { I , {1}, {2}, {3}, {1,2} {1, 3}, {2, 3}, {1, 2, 3}}
For example, if A = {a, b, c, d) and B = {c, d, e, f}, then
Clearly, if A has n elements, then its power set P (A) contains A B = {c, d}.
exactly 2n elements.
SETS, RELATIONS & FUNCTIONS
SETS
1. SET 2.2 Set-Builder Form
A set is a collection of well-defined and well distinguished In this form, we write a variable (say x) representing any
objects of our perception or thought. member of the set followed by a property satisfied by each
member of the set.
1.1 Notations
For example, the set A of all prime numbers less than 10 in
The sets are usually denoted by capital letters A, B, C, etc. the set-builder form is written as
and the members or elements of the set are denoted by lower- A = {x | x is a prime number less that 10}
case letters a, b, c, etc. If x is a member of the set A, we write The symbol '|' stands for the words 'such that'. Sometimes,
x A (read as 'x belongs to A') and if x is not a member of the we use the symbol ':' in place of the symbol '|'.
set A, we write x A (read as 'x does not belong to A,). If x
and y both belong to A, we write x, y A. 3. TYPES OF SETS
2. REPRESENTATION OF A SET
3.1 Empty Set or Null Set
Usually, sets are represented in the following two ways :
A set which has no element is called the null set or empty
(i) Roster form or Tabular form
set. It is denoted by the symbol I .
(ii) Set Builder form or Rule Method
2.1 Roster Form For example, each of the following is a null set :
(a) The set of all real numbers whose square is –1.
In this form, we list all the member of the set within braces
(curly brackets) and separate these by commas. For example, (b) The set of all rational numbers whose square is 2.
the set A of all odd natural numbers less that 10 in the Roster (c) The set of all those integers that are both even and odd.
form is written as :
A set consisting of atleast one element is called a
A = {1, 3, 5, 7, 9}
non-empty set.
3.2 Singleton Set
A set having only one element is called singleton set.
For example, {0} is a singleton set, whose only member is 0.
(i) In roster form, every element of the set is listed
only once. 3.3 Finite and Infinite Set
(ii) The order in which the elements are listed is
A set which has finite number of elements is called a finite
immaterial.
set. Otherwise, it is called an infinite set.
For example, each of the following sets denotes
For example, the set of all days in a week is a finite set
the same set {1, 2, 3}, {3, 2, 1}, {1, 3, 2}
whereas the set of all integers, denoted by
{............ -2, -1, 0, 1, 2,...} or {x | x is an integer}, is an infinite set.
An empty set I which has no element in a finite set A is
called empty of void or null set.
, SETS, RELATIONS & FUNCTIONS
3.4 Cardinal Number 4. OPERATIONS ON SETS
The number of elements in finite set is represented by n(A), 4.1 Union of Two Sets
known as Cardinal number.
3.5 Equal Sets The union of two sets A and B, written as A B (read as 'A
union B'), is the set consisting of all the elements which are
Two sets A and B are said to be equals, written as A = B, if
every element of A is in B and every element of B is in A. either in A or in B or in both Thus,
3.6 Equivalent Sets A B = {x : x A or x B}
Clearly, x A B x A or x B, and
Two finite sets A and B are said to be equivalent, if n
(A) = n (B). Clearly, equal sets are equivalent but equivalent x A B x A and x B.
sets need not be equal.
For example, the sets A = { 4, 5, 3, 2} and B = {1, 6, 8, 9} are
equivalent but are not equal.
3.7 Subset
Let A and B be two sets. If every elements of A is an element
of B, then A is called a subset of B and we write A B or
B A (read as 'A is contained in B' or B contains A'). B is
called superset of A.
For example, if A = {a, b, c, d} and B = {c, d, e, f}, then
A B = {a, b, c, d, e, f}
4.2 Intersection of Two sets
(i) Every set is a subset and a superset itself.
The intersection of two sets A and B, written as A B
(ii) If A is not a subset of B, we write A B.
(read as ‘A’ intersection ‘B’) is the set consisting of all the
(iii) The empty set is the subset of every set. common elements of A and B. Thus,
(iv) If A is a set with n(A) = m, then the number of A B = {x : x A and x B}
subsets of A are 2m and the number of proper
subsets of A are 2m -1. Clearly, x A B x A and x B, and
For example, let A = {3, 4}, then the subsets of A x A B x A or x B.
are I , {3}, {4}. {3, 4}. Here, n(A) = 2 and number
of subsets of A = 22 = 4. Also, {3} {3,4}and {2,3}
{3, 4}
3.8 Power Set
The set of all subsets of a given set A is called the power set
of A and is denoted by P(A).
For example, if A = {1, 2, 3}, then
P(A) = { I , {1}, {2}, {3}, {1,2} {1, 3}, {2, 3}, {1, 2, 3}}
For example, if A = {a, b, c, d) and B = {c, d, e, f}, then
Clearly, if A has n elements, then its power set P (A) contains A B = {c, d}.
exactly 2n elements.
