Chapter 1 Sets Class Notes
Set
A set is a well-defined collection of objects.
Representation of Sets
There are two methods of representing a set
• Roster or Tabular form In the roster form, we list all the members of the set within braces { } and separate by
commas.
• Set-builder form In the set-builder form, we list the property or properties satisfied by all the elements of the
sets.
Types of Sets – Class 11 Maths Notes
• Empty Sets: A set which does not contain any element is called an empty set or the void set or null set and it
is denoted by {} or Φ.
• Singleton Set: A set consists of a single element, is called a singleton set.
• Finite and infinite Set: A set which consists of a finite number of elements, is called a finite set, otherwise the
set is called an infinite set.
• Equal Sets: Two sets A and 6 are said to be equal, if every element of A is also an element of B or vice-versa,
i.e. two equal sets will have exactly the same element.
• Equivalent Sets: Two finite sets A and 6 are said to be equal if the number of elements are equal, i.e. n(A) =
n(B)
Subset –
A set A is said to be a subset of set B if every element of set A belongs to set B. In symbols, we write
A ⊆ B, if x ∈ A ⇒ x ∈ B
Note:
• Every set is o subset of itself.
• The empty set is a subset of every set.
• The total number of subsets of a finite set containing n elements is 2n.
Intervals as Subsets of R
Let a and b be two given real numbers such that a < b, then
• an open interval denoted by (a, b) is the set of real numbers {x : a < x < b}.
• a closed interval denoted by [a, b] is the set of real numbers {x : a ≤ x ≤ b}.
• intervals closed at one end and open at the others are known as semi-open or semi-closed interval and
denoted by (a, b] is the set of real numbers {x : a < x ≤ b} or [a, b) is the set of real numbers {x : a ≤ x < b}.
Power Set
The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). If the number of
elements in A i.e. n(A) = n, then the number of elements in P(A) = 2n.
Universal Set
A set that contains all sets in a given context is called the universal set.
Venn-Diagrams
Venn diagrams are the diagrams, which represent the relationship between sets. In Venn-diagrams the
universal set U is represented by point within a rectangle and its subsets are represented by points in closed
curves (usually circles) within the rectangle.
Operations of Sets
Union of sets: The union of two sets A and B, denoted by A ∪
B is the set of all those elements which are either
in A or in B or in both A and B. Thus, A ∪
B = {x : x ∈
A or x B}.∈
Intersection of sets: The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements which are
common to both A and B.
Thus, A ∩ B = {x : x ∈ A and x ∈ B}
Disjoint sets: Two sets A and B are said to be disjoint, if A ∩ B = Φ.
Set
A set is a well-defined collection of objects.
Representation of Sets
There are two methods of representing a set
• Roster or Tabular form In the roster form, we list all the members of the set within braces { } and separate by
commas.
• Set-builder form In the set-builder form, we list the property or properties satisfied by all the elements of the
sets.
Types of Sets – Class 11 Maths Notes
• Empty Sets: A set which does not contain any element is called an empty set or the void set or null set and it
is denoted by {} or Φ.
• Singleton Set: A set consists of a single element, is called a singleton set.
• Finite and infinite Set: A set which consists of a finite number of elements, is called a finite set, otherwise the
set is called an infinite set.
• Equal Sets: Two sets A and 6 are said to be equal, if every element of A is also an element of B or vice-versa,
i.e. two equal sets will have exactly the same element.
• Equivalent Sets: Two finite sets A and 6 are said to be equal if the number of elements are equal, i.e. n(A) =
n(B)
Subset –
A set A is said to be a subset of set B if every element of set A belongs to set B. In symbols, we write
A ⊆ B, if x ∈ A ⇒ x ∈ B
Note:
• Every set is o subset of itself.
• The empty set is a subset of every set.
• The total number of subsets of a finite set containing n elements is 2n.
Intervals as Subsets of R
Let a and b be two given real numbers such that a < b, then
• an open interval denoted by (a, b) is the set of real numbers {x : a < x < b}.
• a closed interval denoted by [a, b] is the set of real numbers {x : a ≤ x ≤ b}.
• intervals closed at one end and open at the others are known as semi-open or semi-closed interval and
denoted by (a, b] is the set of real numbers {x : a < x ≤ b} or [a, b) is the set of real numbers {x : a ≤ x < b}.
Power Set
The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). If the number of
elements in A i.e. n(A) = n, then the number of elements in P(A) = 2n.
Universal Set
A set that contains all sets in a given context is called the universal set.
Venn-Diagrams
Venn diagrams are the diagrams, which represent the relationship between sets. In Venn-diagrams the
universal set U is represented by point within a rectangle and its subsets are represented by points in closed
curves (usually circles) within the rectangle.
Operations of Sets
Union of sets: The union of two sets A and B, denoted by A ∪
B is the set of all those elements which are either
in A or in B or in both A and B. Thus, A ∪
B = {x : x ∈
A or x B}.∈
Intersection of sets: The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements which are
common to both A and B.
Thus, A ∩ B = {x : x ∈ A and x ∈ B}
Disjoint sets: Two sets A and B are said to be disjoint, if A ∩ B = Φ.
