Sets and Relations
Set
Set is a collection of well defined objects which are distinct from each
other. Sets are usually denoted by capital letters A, B, C ,K and
elements are usually denoted by small letters a , b, c,... .
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 a member of A. If a does not belongs to A, we write
a ∉ A.
Standard Notations
N : A set of all natural numbers.
W : A set of all whole numbers.
Z : A set of all integers.
Z+ / Z− : A set of all positive/negative integers.
Q : A set of all rational numbers.
Q+ / Q− : A set of all positive/negative rational numbers.
R : A set of all real numbers.
R+ / R− : A set of all positive/negative real numbers.
C : A set of all complex numbers.
Methods for Describing a Set
(i) Roster Form / Listing Method / Tabular Form In this
method, a set is described by listing the elements, separated by
commas and enclosed within braces.
e.g. If A is the set of vowels in English alphabet, then
A = { a , e, i , o , u }
(ii) Set Builder Form / Rule Method In this method, we write
down a property or rule which gives us all the elements of the set.
e.g. A = { x : x is a vowel in English alphabet}
Types of Sets
(i) Empty/Null/Void Set A set containing no element, it is denoted
by φ or { }.
, (ii) Singleton Set A set containing a single element.
(iii) Finite Set A set containing finite number of elements or no
element.
Note : Cardinal Number (or Order) of a Finite Set The number of
elements in a given finite set is called its cardinal number. If A is a finite
set, then its cardinal number is denoted by n ( A).
(iv) Infinite Set A set containing infinite number of elements.
(v) Equivalent Sets Two sets are said to be equivalent, if they
have same number of elements.
If n( A) = n( B), then A and B are equivalent sets.
(vi) Equal Sets Two sets A and B are said to be equal, if every
element of A is a member of B and every element of B is a member
of A and we write it as A = B.
Subset and Superset
Let A and B be two sets. If every element of A is an element of B, then
A is called subset of B and B is called superset of A and written as
A ⊆ B or B ⊇ A.
Power Set
The set formed by all the subsets of a given set A, is called power set of
A, denoted by P ( A).
Universal Set (U)
A set consisting of all possible elements which occurs under
consideration is called a universal set.
Proper Subset
If A is a subset of B and A ≠ B, then A is called proper subset of B and
we write it as A ⊂ B.
Comparable Sets
Two sets A and B are comparable, if A ⊆ B or B ⊆ A.
Non-comparable Sets
For two sets A and B, if neither A ⊆ B nor B ⊆ A, then A and B are
called non-comparable sets.
Disjoint Sets
Two sets A and B are called disjoint, if A ∩ B = φ . i.e. they do not have
any common element.
Set
Set is a collection of well defined objects which are distinct from each
other. Sets are usually denoted by capital letters A, B, C ,K and
elements are usually denoted by small letters a , b, c,... .
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 a member of A. If a does not belongs to A, we write
a ∉ A.
Standard Notations
N : A set of all natural numbers.
W : A set of all whole numbers.
Z : A set of all integers.
Z+ / Z− : A set of all positive/negative integers.
Q : A set of all rational numbers.
Q+ / Q− : A set of all positive/negative rational numbers.
R : A set of all real numbers.
R+ / R− : A set of all positive/negative real numbers.
C : A set of all complex numbers.
Methods for Describing a Set
(i) Roster Form / Listing Method / Tabular Form In this
method, a set is described by listing the elements, separated by
commas and enclosed within braces.
e.g. If A is the set of vowels in English alphabet, then
A = { a , e, i , o , u }
(ii) Set Builder Form / Rule Method In this method, we write
down a property or rule which gives us all the elements of the set.
e.g. A = { x : x is a vowel in English alphabet}
Types of Sets
(i) Empty/Null/Void Set A set containing no element, it is denoted
by φ or { }.
, (ii) Singleton Set A set containing a single element.
(iii) Finite Set A set containing finite number of elements or no
element.
Note : Cardinal Number (or Order) of a Finite Set The number of
elements in a given finite set is called its cardinal number. If A is a finite
set, then its cardinal number is denoted by n ( A).
(iv) Infinite Set A set containing infinite number of elements.
(v) Equivalent Sets Two sets are said to be equivalent, if they
have same number of elements.
If n( A) = n( B), then A and B are equivalent sets.
(vi) Equal Sets Two sets A and B are said to be equal, if every
element of A is a member of B and every element of B is a member
of A and we write it as A = B.
Subset and Superset
Let A and B be two sets. If every element of A is an element of B, then
A is called subset of B and B is called superset of A and written as
A ⊆ B or B ⊇ A.
Power Set
The set formed by all the subsets of a given set A, is called power set of
A, denoted by P ( A).
Universal Set (U)
A set consisting of all possible elements which occurs under
consideration is called a universal set.
Proper Subset
If A is a subset of B and A ≠ B, then A is called proper subset of B and
we write it as A ⊂ B.
Comparable Sets
Two sets A and B are comparable, if A ⊆ B or B ⊆ A.
Non-comparable Sets
For two sets A and B, if neither A ⊆ B nor B ⊆ A, then A and B are
called non-comparable sets.
Disjoint Sets
Two sets A and B are called disjoint, if A ∩ B = φ . i.e. they do not have
any common element.
