CBSE Important Question of RELATIONS AND FUNCTIONS Class 12 Maths

MCQ:

I. If a relation R on the set {1,2,3,4] be defined by {(1,2)}, then R is

a) Reflexive                  b) Transitive      c) Symmetric                d) None of these

2. Which of the following functions from Z to Z are one-one and onto?

a) f(x) = x³                 b) f(x) = x+2          c) f(x)= 2x + I          d) f(x) = x² + 1

3. The function f: R → R given by f(x) = cos x, x ∈ R is

a) one-one but not onto               b) onto but not one-one

c) one-one and onto             d) neither one-one nor onto

4. Greatest integer function f(x) = [x]  is

a) one-one          b) many-one           c) both (a) & (b)        d) none of these

• The maximum number of equivalence relations on the set A = {1, 2, 3} are

a) 1  b) 2     c) 3 d) 5

• If R = {(x, y) : x² + y2=4 ,x,y ∈ Z} is a relation of Z, then the domain of R is

a) {0,1,2}  b) {-2,0,2}        c) {-2,-1,1,2} d) {1,2}

• Consider the non-empty set consisting of children in a family and a relation R defined as a R b if a is sister of b. Then R is

a) Symmetric but not transitive                       b) transitive but not symmetric

c) Neither symmetric nor transitive           d) both symmetric and transitive

• Number of relations that can be defined on the set A = {a, b, c, d} is

a) 2³         b) 4⁴     c) 4²     d) 2¹⁶

9. Let R be the relation in the set Z of all integers defined by R = {(x, y): x—y is an integer}. Then R is

a) Reflexive          b) Transitive               c) Symmetric       d) an equivalence relation

10.Let S be the set of all real numbers. Then the relation R = {(a, b): 1 + ab > 0} on S is

a) reflexive, symmetric but not transitive b) reflexive, transitive but not symmetric

c) reflexive, symmetric and transitive              d) both symmetric and transitive but not reflexive

Assertion —reasoning:

1. Both A and R are true and R is the correct explanation of A
2. Both A and R are true and R is not the correct explanation of A
3. A is true but R is false
4. A is false but R is true.

Il. Assertion(A): a relation R ={la-bl< 2 } defined on the set A ={1,2,3,4,5} is reflexive.

Reason(R): A relation R on the set A is said to reflexive if for (a,b) ∈ R and (b,c) ∈ R we have (a,c) ∈ R

1. Assertion(A): Let A= {2,4,6} , B={3,5,7,9} and defined a function f = {(2,3),(4,5), (6,7)} from A to B ,then f is not onto.

Reason(R): A function f: B is said to be onto, if every element of B is the image of some element of A under f.

1. Assertion(A): The smallest integer function f(x) is one-one.

Reason(R): A function is one-one if f(x) = f(y) ⇒x = y.

14. Assertion(A): The function f:      R, f(x) = ׀x׀ is not one-one.

Reason(R): The function f(x) = ׀x׀ is not onto.

Long questions:

1. If f : N → N be the function defined by f(x) = 4x³ +7, check whether f is a one-one and onto function or not.
2. show that the function f: R → { x∈ R : -1 < x <1 } defined by f(x) = , x∈ R is one-one and onto function.
3. A function f: [-4,4] → [0,4] , given by . Show that f is an onto function but not one-one. Further find all possible values of a, f(a) =

1. Consider  given by . show that f is one-one and onto .

3

1. Let A = {1,2,3,…………9}and the relation R on the set A×A defined by (a,b) R (c,d) ↔a +d = b +c for all (a,b),(c,d) ∈ A ×A. Prove that R is an equivalence relation. Also find [(2,5)].
2. Show that the relation R on the set A = {x ∈ Z; 0≤x≤ 12} , given by R = { (a,b) : ׀a – b׀ is a multiple of 4} is an equivalence relation. Find the set of all elements related to 1.

Practice questions:

I. Let A = R — {3} and B = R- {1}. Find the value of a such that the function f:A→ B defined by is onto. Also , check whether the given function is one-one or not.

• Let N denote the set of all natural numbers and R be the relation on N × N defined by

(a , b) R (c,d) imply that ad(b+c) = bc(a+d).

Check whether R is an equivalence relation or not on N × N.

• Show that the relation R on 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 points related to a point P≠ (0,0) is the circle passing through P with origin as centre.
•  Show that f : N -+ N defined by

is many-one and onto function.

• Test whether the relation R on Z defined by R = {(a,b): ׀a—b׀ ≤ 5} is reflexive, symmetric, and transitive.
• Show that the relation R in the set A={1,2,3,4,5}, given by R = {(a,b) : ׀a —b׀ is divisible by 2}, 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 elements of {1,3,5} is related to any element of {2,4}.
• Let R be the equivalence relation in the set A = {0,1,2,3,4,5} given by R (a,b): 2 divides (a — b)}. Write the equivalence class [0].
• Let R = {(a, a³ ): a is a prime number less than 5} be a relation. Find the range of R.
• Check whether the function f : N → N given by f(x) = 9x² + 6x-5 is one-one and onto or not.

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

ll. Check whether a function defined as is one-one and onto or not.

12. Let L be the set of all lines in XY plane and R be the relation on L defined as

R = { (L_l,L₂): L_l is parallel to L₂} .

Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

13. Prove that a function f: (0,∞)→ [-5, ∞) defined as f(x) = 4x² + 4x— 5 is both one-one and onto.

1. Let f:R →R   given by f(x) =. show that f is one-one and onto

1. Show that the signum function f: R→ R, given by, f(x) = neither one-one nor onto.