# Relations and Functions - Online Test

Q1. The domain of the function √cosx-1
Explaination / Solution:

This function exists only if cosx−1≥0 ⇒cosx≥1 ⇒cosx>1 OR cosx=1 since maximum value of cosine function is 1 so, cos x >1 is not possible so, cosx=1 cosx=cos2nπ ⇒x=2nπ(n∈I)
Q2. If R is a relation from a non – empty set A to a non – empty set B, then
Explaination / Solution:

Let A and B be two sets . Then a relation R from set A to set B is a subset of A×B.Thus , R is a relation from A to B ⇔R⊆A×B⇔R⊆A×B.

Q3. Let A = { 2 , 3 , 6 }. Which of the following relations on A are reflexive ?
Explaination / Solution:

is a reflexive on A , because ( a,a ) ∈ R1  for each  a ∈ A

Q4. Let R be the relation on N defined as x R y if x + 2 y = 8. The domain of R is
Explaination / Solution:

As x R y if x + 2 y = 8 , therefore , domain of the relation R is given by x = 8 – 2y∈N. When y = 1, ⇒x = 6 ,when y = 2, ⇒x =4 , when y =3 , ⇒x = 2 . therefore domain is { 2, 4, 6 }.

Q5. Which of the following is not an equivalence relation on I, the set of integers ; x, y
Explaination / Solution:

f R is a relation defined by xRy:ifx⩽y, then R is reflexive and transitive .But , it is not symmetric. Hence , R is not an equivalence relation.

Q6. Let A = {a, b, c} and R = {(a, a), (b, b), (c, c), (b, c) } be a relation on A. Here, R is
Explaination / Solution:

Any relation R is reflexive if x R x for all x ∈ R. Here ,(a, a), (b, b), (c, c)∈ R. Therefore , R is reflexive.

Q7. R = {(1, 1), (2, 2), (1, 2), (2, 1), (2, 3)} be a relation on A, then R is
Explaination / Solution:

A relation R on a non empty set A is said to be reflexive if xRx for all x R , Therefore , R is not reflexive.
A relation R on a non empty set A is said to be symmetric if xRyyRx, for all x , y R Therefore, R is not symmetric.
A relation R on a non empty set A is said to be antisymmetric if xRy and yRxx = y , for all x , y R.Therefore, R is not antisymmetric.

Q8. Let A = {1, 2, 3}. Which of the following is not an equivalence relation on A ?
Explaination / Solution:

A relation R on a non empty set A is said to be reflexive iff xRx for all x  R . A relation R on a non empty set A is said to be symmetric iff xRyyRx, for all x , y R .
A relation R on a non empty set A is said to be transitive iff xRy and yRzxRz, for all x  R. An equivalence relation satisfies all these three properties. .
None of the given relations satisfies all three properties of equivalence relation.

Q9. Let A = {1, 2, 3}, then the relation R = {(1, 1), (2, 2), (1, 3)} on A is
Explaination / Solution:

The given relation is not reflexive , as (3,3)∉R, The given relation is not symmetric , as (1,3)∈ R , but (3,1) ∉R, , The given relation is transitive as (1,1) )∈ R and (1,3) )∈R.

Q10. Let A = {1, 2, 3}, then the relation R = {(1, 1), (1, 2), (2, 1)} on A is
Explaination / Solution:

A relation R on a non empty set A is said to be symmetric iff xRy⇔yRx, for all x , y ∈R Clearly , (1, 2), and (2, 1) both lies in R. Therefore ,R is symmetric.