Let $A =\left \{ 1,2,3 \right \}$
$A\times A ={ (1,1) (2,2) (3,3) (1,2) (2,1) (2,3) (3,2) (1,3) (3,1) }$
A relation $R$ on set $A$ is said to be Equivalence relation if $R$ is
(1)Reflexive (2)Symmetric (3)Transitive
For all Equivalence relation, all $(x R x) \forall A \varepsilon A$ should be present which is Reflexive Relation property.
For Equivalence relation at least ${ (1,1) (2,2) (3,3) }$ elements should be present.
Now For remaining elements ${ (1,2) (2,1) (2,3) (3,2) (1,3) (3,1) }$
According to Symmetric Relation if $(xRy)$ then $(yRx) \forall x,y\varepsilon A \\$
Hence, for all 3 pairs, $(1,2) (2,1) , (2,3) (3,2) ,(1,3) (3,1)$we have 2 choices either it can include or it can't include.
Number of relation of these type=$2^{3}=8$
$R_{1}=\left \{ (1,1)(2,2)(3,3) \right \} \ \ \ R{\color{Green} \checkmark} \ \ \ S {\color{Green} \checkmark} \ \ \ T{\color{Green} \checkmark}$
$R_{2}=\left \{ (1,1)(2,2)(3,3)(1,2) (2,1)\right \}\ \ \ R{\color{Green} \checkmark} \ \ \ S {\color{Green} \checkmark} \ \ \ T{\color{Green} \checkmark}$
$R_{3}=\left \{ (1,1)(2,2)(3,3) (1,3)(3,1)\right \}\ \ \ R{\color{Green} \checkmark} \ \ \ S {\color{Green} \checkmark} \ \ \ T{\color{Green} \checkmark}$
$R_{4}=\left \{ (1,1)(2,2)(3,3) (2,3)(3,2)\right \}\ \ \ R{\color{Green} \checkmark} \ \ \ S {\color{Green} \checkmark} \ \ \ T{\color{Green} \checkmark}$
$R_{5}=\left \{ (1,1)(2,2)(3,3)(1,2) (2,1) (1,3)(3,1)\right \}\ \ \ R{\color{Green} \checkmark} \ \ \ S {\color{Green} \checkmark} \ \ \ T{\color{Red} \times}$
$R_{6}=\left \{ (1,1)(2,2)(3,3) (1,2)(2,1)(2,3)(3,2)\right \}\ \ \ R{\color{Green} \checkmark} \ \ \ S {\color{Green} \checkmark} \ \ \ T{\color{Red} \times}$
$R_{7}=\left \{ (1,1)(2,2)(3,3) (1,3)(3,1)(2,3)(3,2)\right \}\ \ \ R{\color{Green} \checkmark} \ \ \ S {\color{Green} \checkmark} \ \ \ T{\color{Red} \times}$
$R_{8}=\left \{ (1,1)(2,2)(3,3)(1,2) (2,1) (1,3)(3,1)(2,3)(3,2)\right \}\ \ \ R{\color{Green} \checkmark} \ \ \ S {\color{Green} \checkmark} \ \ \ T{\color{Green} \checkmark}$
Here we can see relation $R_{5}$,$R_{6}$ and $R_{7}$ are failed to satisfy the properties of transitive relation.
Hence, total number of equivalence relation=5
Option(B) 5 is the correct choice.