Shortcut Methods
JEE Main

Number of elements in the power set of a set with n elements: 2^n
 Shortcut: The number of elements in the power set of a set with n elements is equal to the total number of subsets of that set, which is 2^n.
 Example: If a set A = {1, 2, 3}, then the power set of A is P(A) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. Thus, the number of elements in the power set of A is 2^3 = 8.

Number of subsets of a set with n elements: 2^n  1
 Shortcut: The number of subsets of a set with n elements is equal to the total number of elements in the power set of that set minus the empty set, which is 2^n  1.
 Example: If a set A = {1, 2, 3}, then the number of subsets of A is 2^3  1 = 8  1 = 7.

Number of ordered pairs of elements from a set with n elements: n*(n1)/2
 Shortcut: The number of ordered pairs of elements from a set with n elements is equal to the number of combinations of n elements taken 2 at a time, which is n*(n1)/2.
 Example: If a set A = {1, 2, 3}, then the number of ordered pairs of elements from A is 3*(31)/2 = 3*2/2 = 3.

Number of unordered pairs of elements from a set with n elements: n*(n1)/2
 Shortcut: The number of unordered pairs of elements from a set with n elements is equal to the number of combinations of n elements taken 2 at a time, which is n*(n1)/2.
 Example: If a set A = {1, 2, 3}, then the number of unordered pairs of elements from A is 3*(31)/2 = 3*2/2 = 3.

Number of relations from a set with m elements to a set with n elements: m^n
 Shortcut: The number of relations from a set with m elements to a set with n elements is equal to the total number of functions from the set with m elements to the set with n elements, which is m^n.
 Example: If a set A = {1, 2} and a set B = {3, 4}, then the number of relations from A to B is 2^4 = 16.

Number of functions from a set with m elements to a set with n elements: n^m
 Shortcut: The number of functions from a set with m elements to a set with n elements is equal to the total number of mappings from the set with m elements to the set with n elements, which is n^m.
 Example: If a set A = {1, 2} and a set B = {3, 4}, then the number of functions from A to B is 4^2 = 16.
CBSE Board Exams

Venn diagrams to represent sets and their relationships
 Shortcut: Draw a rectangle to represent the universal set, and then draw circles or ovals within the rectangle to represent the different sets. The intersection of two sets is the region where the circles or ovals overlap, and the union of two sets is the region covered by both circles or ovals.
 Example: If we have two sets, A = {1, 2, 3} and B = {2, 3, 4}, we can represent them using a Venn diagram as follows: [Image of a Venn diagram with two circles representing sets A and B]
 The intersection of A and B is the set {2, 3}, which is the region where the two circles overlap.
 The union of A and B is the set {1, 2, 3, 4}, which is the region covered by both circles.

Operations on sets, such as union, intersection, and complement
 Shortcut: Remember the symbols for each operation:
 Union: A ∪ B (read as “A union B” or “A or B”)
 Intersection: A ∩ B (read as “A intersection B” or “A and B”)
 Complement: A’ or A^{C} (read as “A complement” or “the complement of A”)
 Example: If A = {1, 2, 3} and B = {2, 3, 4}, then:
 A ∪ B = {1, 2, 3, 4}
 A ∩ B = {2, 3}
 A’ = {4}

Properties of sets, such as commutativity, associativity, and distributivity
 Shortcut: Remember the following properties:
 Commutativity: A ∪ B = B ∪ A and A ∩ B = B ∩ A
 Associativity: (A ∪ B) ∪ C = A ∪ (B ∪ C) and (A ∩ B) ∩ C = A ∩ (B ∩ C)
 Distributivity: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
 Example: If A = {1, 2, 3}, B = {2, 3, 4}, and C = {3, 4, 5}, then:
 (A ∪ B) ∪ C = {1, 2, 3, 4, 5} = A ∪ (B ∪ C)
 (A ∩ B) ∩ C = {3} = A ∩ (B ∩ C)

Cartesian product of sets
 Shortcut: The Cartesian product of two sets A and B is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B.
 Example: If A = {1, 2} and B = {3, 4}, then the Cartesian product of A and B is A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.

Relations and functions, including domain, range, and inverse relations
 Shortcut: Remember the definitions of these concepts:
 A relation from a set A to a set B is a subset of A × B.
 The domain of a relation R is the set of all elements in A that appear as the first element of any ordered pair in R.
 The range of a relation R is the set of all elements in B that appear as the second element of any ordered pair in R.
 The inverse of a relation R is the relation from B to A that consists of all ordered pairs (b, a) such that (a, b) is in R.
 Example: If R = {(1, 3), (2, 4), (3, 5)} is a relation from A = {1, 2, 3} to B = {3, 4, 5}, then:
 The domain of R is A = {1, 2, 3}.
 The range of R is B = {3, 4, 5}.
 The inverse of R is R^{1} = {(3, 1), (4, 2), (5, 3)}.