# Types of Relations

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• Q1

Let X be a family of sets and R be a relation on X defined by 'A is disjoint from B'. Then R is

Marks:1

symmetric.

##### Explanation:
Clearly, the relation is symmetric but it is neither reflexive nor transitive.

• Q2

Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is

Marks:1

not symmetric.

##### Explanation:

(2, 3)  R but (3, 2)  R

R is not symmetric

• Q3

Let R be a relation on a set A such that R = R-1, then R is

Marks:1

symmetric.

##### Explanation:
Let (a, b)  R. Then,
(a, b)  R  (b, a)  R-1   [by defn of R-1]
(b, a)  R                       [ R = R-1]
So, R is symmetric.
• Q4

Let R = {(2, 3), (3, 4)} be a relation defined on the set A = {1, 2, 3, 4}. The minimum number of ordered pairs required to be added in R so that the new relation becomes an equivalence relation is

Marks:1

8

##### Explanation:

Given R = {(2,3), (3,4)}             ...(i)

To make it reflexive, we add the following ordered pairs in R:

R = {1, 1), (2, 2), (3, 3), (4, 4), (2, 3), (3,4) ...(ii)

Hence 4 ordered pairs are added.

To make it symmetric, we again add the following ordered pairs in R from eqn (ii)

{(1,1), (2,2), (3,3), (4,4), (2,3) (3,2), (3,4) (4,3)} ...(iii)

Hence two more ordered pairs are added.

Finally to make it transitive, we add the following ordered pairs in R from eqn (iii)

{(1,1), (2,2), (3,3), (4,4), (2,3) (3,2), (3,4) (4,3), (2,4), (4,2)}

Hence, two more ordered pairs are added.

Total 8 ordered pairs must be added to make the relation R an equivalence relation.

• Q5

For real number x any y, we write xRy $⇔x-y+\sqrt{2}$ is an irrational number. Then the relation R is

Marks:1

reflexive only

##### Explanation:

, an irrational number.

Hence, xRx $\forall$ x, i.e. R is reflexive.

R cannot be symmetric because if x = $\sqrt{2}$, y = 1, then

Similarly, we may show that R is not transitive.