Show that the relation S in the set given by is divisible by 3} is an equivalence relation.
Reflexive
is defined on the set
clearly, S is reflexive as 3 divides
symmetric
further if then 3 divides therefore 3 divides as well
Hence which follows that S is symmetric
Transitive
similarly, if and then and are both divisible by 3.
that is i.e
and i.e
so
is divisible by 3
This shows that S is transitive as
Then S is an equivalence relation in the set A