# Mark the correct alternative in the following question:

Question:

Mark the correct alternative in the following question:

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

(a) 1

(b) 2

(c) 3

(d) 5

Solution:

Consider the relation $R_{1}=\{(1,1)\}$

It is clearly reflexive, symmetric and transitive

Similarly, $R_{2}=\{(2,2)\}$ and $R_{3}=\{(3,3)\}$ are reflexive, symmetric and transitive

Also, $R_{4}=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}$

It is reflexive as $(a, a) \in R_{4}$ for all $a \in\{1,2,3\}$

It is symmetric as $(a, b) \in R_{4} \Rightarrow(b, a) \in R_{4}$ for all $a \in\{1,2,3$

Also, it is transitive as $(1,2) \in R_{4},(2,1) \in R_{4} \Rightarrow(1,1) \in R_{4}$

The relation defined by $R_{5}=\{(1,1),(2,2),(3,3),(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)\}$ is reflexive, symmetric and transitive as well.

symmetric and transitive as well.

Thus, the maximum number of equivalence relation on set $A=\{1,2,3\}$ is 5 .

Hence, the correct alternative is option (d).