Grade XI
Mathematics
Formula
Notes
,#GrowWithGreen
, Formula Notes
● Some important points on sets:
(a) Every set is a subset of itself.
(b) A subset which is not a proper subset is called an improper subset. If A and B are two
equal sets, then A and B are improper subsets of each other.
(c) Every set has only one improper subset and that is itself.
(d) An empty set is a subset of every set.
(e) An empty set is a proper subset of every set except itself.
(f) Every set is a subset of the universal set.
(g) If X ⊆ Y and Y ⊆ X, then X = Y
● If cardinal number of the set A is m, i.e., n (A) = m, then
The number of subsets of A = 2m
The number of proper subsets of A = 2m − 1
● If the number of elements in set A is m, then the number of elements in the power set of A
is 2m.
i.e., nP(A) = 2m, where n(A) = m
● Following are the properties of union of two sets:
1. A ∪ B = B ∪ A
2. A ∪ Φ = A
3. A ∪ A = A
4. (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative law)
5. U ∪ A = U (Law of universal set, U)
● Following are the intersection of two sets are given as follows:
○ A∩B=B∩A
○ Φ∩A=Φ
○ A∩A=A
○ (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative law)
○ U∩A=A (Law of universal set, U)
● n (A∪B) = n (A) + n (B) −n (A∩B)
● If A and B are two disjoint sets i.e., A ∩ B = Φ, then n (A ∩ B) = 0
So, n (A∪B) = n (A) + n (B)
● When the sets A and B are overlapping, the Venn diagram representing A ∪ B can be
shown as:
, Formula
● When the sets A and B are overlapping, the set A ∩ B is the shaded portion of the following
the Venn diagram.
● When the sets A and B are disjoint, the Venn diagrams representing A ∪ B can be shown
as:
● When the sets A and B are disjoint, the Venn diagrams representing A ∩ B can be shown as:
Mathematics
Formula
Notes
,#GrowWithGreen
, Formula Notes
● Some important points on sets:
(a) Every set is a subset of itself.
(b) A subset which is not a proper subset is called an improper subset. If A and B are two
equal sets, then A and B are improper subsets of each other.
(c) Every set has only one improper subset and that is itself.
(d) An empty set is a subset of every set.
(e) An empty set is a proper subset of every set except itself.
(f) Every set is a subset of the universal set.
(g) If X ⊆ Y and Y ⊆ X, then X = Y
● If cardinal number of the set A is m, i.e., n (A) = m, then
The number of subsets of A = 2m
The number of proper subsets of A = 2m − 1
● If the number of elements in set A is m, then the number of elements in the power set of A
is 2m.
i.e., nP(A) = 2m, where n(A) = m
● Following are the properties of union of two sets:
1. A ∪ B = B ∪ A
2. A ∪ Φ = A
3. A ∪ A = A
4. (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative law)
5. U ∪ A = U (Law of universal set, U)
● Following are the intersection of two sets are given as follows:
○ A∩B=B∩A
○ Φ∩A=Φ
○ A∩A=A
○ (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative law)
○ U∩A=A (Law of universal set, U)
● n (A∪B) = n (A) + n (B) −n (A∩B)
● If A and B are two disjoint sets i.e., A ∩ B = Φ, then n (A ∩ B) = 0
So, n (A∪B) = n (A) + n (B)
● When the sets A and B are overlapping, the Venn diagram representing A ∪ B can be
shown as:
, Formula
● When the sets A and B are overlapping, the set A ∩ B is the shaded portion of the following
the Venn diagram.
● When the sets A and B are disjoint, the Venn diagrams representing A ∪ B can be shown
as:
● When the sets A and B are disjoint, the Venn diagrams representing A ∩ B can be shown as:
