Question

Total number of equivalence relations defined in the set \[S = \{ a,b,c\} \] is ?
A) $5$
B) $3!$
C) ${2^3}$
D) ${3^3}$

Answer 1

Hint:
A relation is called an equivalence relation if it is reflexive, symmetric and transitive. The given set contains three elements. The identity relation is always an equivalence relation. Also we can consider other equivalence relations by the definition itself.

Complete step by step solution:
The given set is \[S = \{ a,b,c\} \].
A relation on a set is a subset of the Cartesian product $S \times S$.
A relation is called an equivalence relation if it is reflexive, symmetric and transitive.
A relation is reflexive for every $a \in S$, we have $aRa$.
A relation is called symmetric if for every $a,b \in S$, we have $aRb \Rightarrow bRa$.
A relation is transitive if for every $a,b,c \in S$, $aRb,bRc \Rightarrow aRc$.
The identity relation $\{ (a,a),(b,b),(c,c)\} $ is an equivalence relation, since it is reflexive, symmetric and transitive.
Every other equivalence relation contains the identity relation.
The relations,
${R_1} = \{ (a,a),(b,b),(c,c),(a,b),(b,a)\} $
${R_2} = \{ (a,a),(b,b),(c,c),(b,c),(c,b)\} $
${R_3} = \{ (a,a),(b,b),(c,c),(a,c),(c,a)\} $
are equivalence relations.
Finally the universal relation ${R_4} = \{ (a,a),(b,b),(c,c),(a,b),(b,a),(b,c),(c,b),(a,c),(c,a)\} $ is also an equivalence relation.
This gives the number of equivalence relations is $5$.

Therefore the answer is option A.

Additional information:
For every equivalence relation, we can consider it as a partition on the set. That is, the set can be written as the union of disjoint, non-empty sets. Here those sets are the three singleton sets $\{ a\} ,\{ b\} $ and $\{ c\} $.

Note:
For any set, identity relation defined on it is always an equivalence relation. Since every element is related to itself, it is reflexive, symmetric and transitive. Also every other equivalence relation contains this.