Question

Let $A = \{ 1,2,3\} $. Then number of equivalence relations containing $(1,2)$ is:
(a) $1$
(b) $2$
(c) $3$
(d) $4$

Answer 1

(Hint: Try to figure out all the possible cases and then construct the required sets.)

We have the given set as $A = \{ 1,2,3\} $
Now, it is given in the question that,
We have to calculate the number of equivalence relations containing $(1,2)$
That is,$1$ is related to $2$.

 So, we have two possible cases:

Case 1: When 1 is not related to 3,
 then the relation
\[{R_1} = \left\{ {\left( {1,1} \right),\left( {1,2} \right),\left( {2,1} \right),\left( {2,2} \right),\left( {3,3} \right)} \right\}\;\] is the only equivalence relation containing $(1,2)$.

Case 2: When 1 is related to 3,
then the relation
 \[A \times A\; = \{ \;\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right),\left( {1,2} \right),\left( {2,1} \right),\left( {1,3} \right),\left( {3,1} \right),\left( {2,3} \right),\left( {3,2} \right)\;\} \] is the only equivalence relation containing $(1,2)$.

∴ There are two equivalence relations on A with the equivalence property.
So, the required solution is (b) 2.


Note: In solving these questions, we must have an understanding of the equivalence, reflexive, symmetric relations, transitive, etc. As we know that, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.