Commerce (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

If R and S are relations on a set A, then prove that R is reflexive and S is any relation ⇒ R ∪ S is reflexive ?

If R and S are transitive relations on a set A, then prove that R ∪ S may not be a transitive relation on A.

Let C be the set of all complex numbers and C₀ be the set of all no-zero complex numbers. Let a relation R on C₀ be defined as

`z_1 R  z_2  ⇔ (z_1 -z_2)/(z_1 + z_2)` is real for all z₁, z₂ ∈ C₀.

Show that R is an equivalence relation.

Write the domain of the relation R defined on the set Z of integers as follows:-
(a, b) ∈ R ⇔ a² + b² = 25

If R = {(x, y) : x² + y² ≤ 4; x, y ∈ Z} is a relation on Z, write the domain of R.

Write the identity relation on set A = {a, b, c}.

Write the smallest reflexive relation on set A = {1, 2, 3, 4}.

If R = {(x, y) : x + 2y = 8} is a relation on N by, then write the range of R.

If R is a symmetric relation on a set A, then write a relation between R and R⁻¹.

Let R = {(x, y) : |x² − y²| <1) be a relation on set A = {1, 2, 3, 4, 5}. Write R as a set of ordered pairs.

If A = {2, 3, 4}, B = {1, 3, 7} and R = {(x, y) : x ∈ A, y ∈ B and x < y} is a relation from A to B, then write R⁻¹.

Let A = {3, 5, 7}, B = {2, 6, 10} and R be a relation from A to B defined by R = {(x, y) : x and y are relatively prime}. Then, write R and R⁻¹.

Define a reflexive relation ?

Define a symmetric relation ?

Define a transitive relation ?

Define an equivalence relation ?

If A = {3, 5, 7} and B = {2, 4, 9} and R is a relation given by "is less than", write R as a set ordered pairs.

A = {1, 2, 3, 4, 5, 6, 7, 8} and if R = {(x, y) : y is one half of x; x, y ∈ A} is a relation on A, then write R as a set of ordered pairs.

Let A = {2, 3, 4, 5} and B = {1, 3, 4}. If R is the relation from A to B given by a R b if "a is a divisor of b". Write R as a set of ordered pairs.

State the reason for the relation R on the set {1, 2, 3} given by R = {(1, 2), (2, 1)} to be transitive ?