Total number of equivalence relations defined in the set $S = \{ a,b,c\} $is ?

Hint: For a set $A$ the equivalence relations will be the subset of set $A \times A$ . There are three types of relationship between the all of the elements of a set and these are reflexive, symmetric and transitive. In the reflexive the elements in the set has same value, and in the symmetric the set follows the relation \[\left( {x\,R\,y} \right)\] implies \[\left( {y\,R\,x} \right)\]. And all the other relations come under transitive.

Complete step-by-step answer:
The reflexive equivalence relation of the given set will be the smallest relation and that is \[{S_1} = \left\{ {\left( {a,a} \right),\left( {b,b} \right),\left( {c,c} \right)} \right\}\]
Now, we can add two ordered pairs of two different elements to get the symmetric equivalence relations \[{S_2} = \left\{ {\left( {a,a} \right),\left( {b,b} \right),\left( {c,c} \right),\left( {a,b} \right),\left( {b,a} \right)} \right\}\]
Similarly we can get ${S_3}$ and ${S_4}$ as symmetric equivalence relation by taking \[\left( {b,c} \right),\left( {c,b} \right)\] , and \[\left( {a,c} \right),\left( {c,a} \right)\] respectively.
\[{S_3} = \left\{ {\left( {a,a} \right),\left( {b,b} \right),\left( {c,c} \right),\left( {b,c} \right),\left( {c,b} \right)} \right\}\]
And
\[{S_4} = \left\{ {\left( {a,a} \right),\left( {b,b} \right),\left( {c,c} \right),\left( {a,c} \right),\left( {c,a} \right)} \right\}\]
Now, we can get the largest equivalence relation by taking all the relations and that is universal equivalence relation
\[{S_5} = \left\{ {\left( {a,a} \right),\left( {b,b} \right),\left( {c,c} \right),\left( {a,b} \right),\left( {b,a} \right),\left( {a,c} \right),\left( {c,a} \right),\left( {b,c} \right),\left( {c,b} \right)} \right\}\]
Therefore, we have the total number of 5 equivalence relations for the given set.

Note: In such types of questions, you should always start from reflexive equivalence relations, then you can move to symmetric and transitive equivalence relations. The universal equivalence consists of all the relations in a set.