PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Let L be the set of all lines in XY-plane and R be the relation in L defined as R = {L₁, L₂) : L₁ is parallel to L₂}. Show that R is an equivalence relation. Find the set of all lines related to the line y= 2x + 4.

Show that the relation R, defined in the set A of all polygons as R = {(P₁, P₂) : P₁ and P₂ have the same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right-angled triangle T with sides 3, 4 and 5?

Let O be the origin. We define a relation between two points P and Q in a plane if OP = OQ. Show that the relation, so defined is an equivalence relation.

Let R be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.

Let S be a relation on the set R of all real numbers defined by
S = {(a, b) ∈ R × R : a² + b² = 1}
Prove that S is not an equivalence relation on R.

Let Z be the set of all integers and Z₀ be the set of all non-zero integers. Let a relation R on Z × Z₀be defined as (a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z₀,
Prove that R is an equivalence relation on Z × Z₀.

If R and S are relations on a set A, then prove that R and S are symmetric ⇒ R ∩ S and R ∪ S are symmetric ?

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 ?