List of top Mathematics Questions on Relations and Functions asked in CBSE CLASS XII

Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b² } is neither reflexive nor symmetric nor transitive.

Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive
but not symmetric. 

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R={(a,b) : Ia-bI is even},  is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of 2, 4}.

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.
  A. R is reflexive and symmetric but not transitive.
  B. R is reflexive and transitive but not symmetric.
  C. R is symmetric and transitive but not reflexive.
  D. R is an equivalence relation. 

Show that the relation R in the set A of points in a plane given by R = {(P, Q): distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

1. f : R→R defined by f(x)=3−4x
2. f : R→R defined by f(x)=1+x2

State whether the following statements are true or false. Justify. 
(i) For an arbitrary binary operation * on a set N, a * a=a ∀ a * N. 
(ii) If * is a commutative binary operation on N, then a * (b * c)= (a * b)* a

Given an example of a relation. Which is
(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Check whether the relation R in R defined as R = {(a, b): a ≤ b³} is reflexive, symmetric or transitive

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive. 

Let A=N×N and * be the binary operation on A defined by (a, b)*(c, d)=(a+c,b+d) Show that * is commutative and associative. Find the identity element for * on A, if any.

Show that the relation R defined in the set A of all triangles as R = {(T₁, T₂): T₁ is similar to T₂}, is equivalence relation. Consider three right angle triangles T₁ with sides 3, 4, 5, T₂ with sides 5, 12, 13 and T₃ with sides 6, 8, 10. Which triangles among T₁, T₂ and T₃ are related?

Show that each of the relation R in the set A = { x ∈ Z : 0 ≤ x ≤ 12}, given by 
 I. R={(a,b):Ia-bI is a multiple of 4} 
 II. R={(a,b):a=b}
is an equivalence relation. Find the set of all elements related to 1 in each case.

Consider f: R₊\(\to\) [−5,∞) given by f(x) = 9x² + 6x − 5. Show that f is invertible with 
\(f^{-1}(y) = \frac {(\sqrt {y+6})-1}{3}\)

Consider f: {1, 2, 3} \(\to\) {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f⁻¹ and show that (f⁻¹)⁻¹= f.

Consider f: R₊\(\to\)[4,∞) given by f(x) = x²+4. Show that f is invertible with the inverse f⁻¹ of given f by \(f^{-1}(y)= \sqrt {y-4}\) , where R₊is the set of all non-negative real numbers.

Consider f: R\(\to\)R given by f(x) = 4x+3. Show that f is invertible. Find the inverse of f.