Relations and Functions - Online Test

Q1. If A = { 1, 2, 3}, then the relation R = {(1, 2), (2, 3), (1, 3)} in A is
Answer : Option B
Explaination / Solution:

A relation R on a non empty set A is said to be transitive if x Ry and y Rz ⇒xRz, for all x ∈ R. Here , (1, 2) and (2, 3) belongs to R implies that (1, 3) belongs to R.

Q2. A relation R from C(complex no.) to R(real no.) is defined by x Ry iff |x| = y. Which of the following is correct?
Answer : Option B
Explaination / Solution:

As  i.e. x is a complex no., then.

Q3. If R is a relation from a set A to a set B and S is a relation from B to C, then the relation S∘R.
Answer : Option C
Explaination / Solution:

Let R and S be two relations from sets A to B and B to C respectively.Then we can define a relation 

from A to C such that   this relation is called the composition of R and S.


Q4. The binary operation * defined on the set of integers as a∗b=|a−b|−1is
Answer : Option D
Explaination / Solution:

Here * is commutative as b*a = |b−a|−1=|a−b|−1=a∗b. Because ,|−x|=|x|for all x∈R.

Q5.

R is a relation from { 11, 12, 13} to {8, 10, 12} defined by y = x – 3. The relation =


Answer : Option B
Explaination / Solution:

R is a relation from { 11, 12, 13} to {8, 10, 12} defined by y = x – 3. The relationis given by x = y + 3,from {8, 10, 12} to { 11, 12, 13}  relation = {(8,11),(10,13)}.

Q6. Given the relation R = {(1, 2), (2, 3)} on the set {1, 2, 3}, the minimum number of ordered pairs which when added to R make it an equivalence relation is =
Answer : Option B
Explaination / Solution:

To make the relation an equivalence relation , the following ordered pairs are required (1,1),(2,2),(3,3)(2,1)(3,2)(1,3),(3,1).

Q7. Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1),(3,2)} be a relations on the set A = {1, 2, 3, 4}. The relation R is
Answer : Option C
Explaination / Solution:

R is said to be symmetric if (a,b)∈R⇒(b,a)∈R ,here (1,3)∈R⇒(3,1)∈R etc.

Q8. In Z , the set of integers , inverse of – 7 , w.r.t. ‘ * ‘ defined by a*b = a +b + 7 for all a,b∈Z ,is
Answer : Option B
Explaination / Solution:

If ‘ e ‘ is the identity ,then a*e = a ⇒a + e + 7 = a ⇒ e = - 7 . Also,inverse of e is e itself. Hence , inverse of -7 is -7.

Q9. A Relation is a
Answer : Option D
Explaination / Solution:

By definition of Relation, : A relation from a non-empty set A to a non empty set B is a subset of A x B . If ( x, y) ∈ R , then we write xRy and we say that x is related to y through R.

Q10. A relation R on a set A is called an empty relation if
Answer : Option C
Explaination / Solution:

For any set A ,an empty relation may be defined on A as: there is no element exists in the relation set which satisfies the relation for a given set A i.e. let A={1,2,3,4,5} and R={(a,b): a,b ∈A and a+b= 10},so we get R={ } which is an empty relation.