Question

# The maximum number of equivalence relations on the set $A=\left\{ 1,2,3 \right\}$ is(a) $1$(b) $2$(c) $3$(d) $5$

Hint: Will find all the possible relations that are equivalence i.e. we will find all the possible relations that are symmetric, reflexive and transitive at the same time.

Before finding the maximum number of equivalence relation on the set $A=\left\{ 1,2,3 \right\}$, we will first discuss what do we mean by the equivalence relation?
A relation is said to be an equivalence relation if it is,
1) Reflexive - A relation $R$ on a set $A$ is said to be reflexive if $\left( a,a \right)$ is there in
relation $R$ $\forall a\in A$.
2) Symmetric – A relation $R$ on a set $A$ is said to be symmetric when, if $\left( a,b \right)$ is
there in the relation, then $\left( b,a \right)$ should also be there in the relation for $a,b\in A$.
3) Transitive – A relation $R$ on a set $A$ is said to be transitive when, if $\left( a,b \right)$ and
$\left( b,c \right)$ are there in the relation, then $\left( a,c \right)$ should also be there in the
relation for $a,b,c\in A$.
For a relation which is defined on the set $A=\left\{ 1,2,3 \right\}$, all possible relations that are
equivalence are,
1) $\left\{ \left( 1,1 \right),\left( 2,2 \right),\left( 3,3 \right) \right\}$
2) $\left\{ \left( 1,1 \right),\left( 2,2 \right),\left( 3,3 \right),\left( 1,2 \right),\left( 2,1 \right) \right\}$
3) $\left\{ \left( 1,1 \right),\left( 2,2 \right),\left( 3,3 \right),\left( 1,3 \right),\left( 3,1 \right) \right\}$
4) $\left\{ \left( 1,1 \right),\left( 2,2 \right),\left( 3,3 \right),\left( 2,3 \right),\left( 3,2 \right) \right\}$
5) $\left\{ \left( 1,1 \right),\left( 2,2 \right),\left( 3,3 \right),\left( 1,2 \right),\left( 2,1 \right),\left( 1,3 \right),\left( 3,1 \right),\left( 2,3 \right),\left( 3,2 \right) \right\}$
All the possible relations on the set $A=\left\{ 1,2,3 \right\}$ that are equivalence are made in the
above list. So, the maximum number of equivalence relations that are possible on the set $A=\left\{ 1,2,3 \right\}$ is equal to 5.
Therefore option (d) is correct answer

Note: There is a possibility that one may make mistakes while writing all the possible equivalence relation that can be formed on the given set $A$. To avoid such mistakes, one can follow these steps. First write down the reflexive relation. Then start writing down the relations that are both reflexive as well as symmetric taking two numbers from set $A$ at a single time. Finally, write down the union relation of all the relations that are generated from the second step.