for the set A={1,2,3} define the relation R in set A as follows:

R={(1,1) (2,2) (3,3) (1,3)}. write the ordered pairs to be added to R to make it the smallest equivalent relation.

R = {(1,1); (2,2); (3,3); (1,3)}

the given relation is reflexive as $aRa$ for $\forall a\in R$

the given relation is not symmetric since for the pair (1,3) (3,1) must be there

so we will add (3,1).

thus R = {(1,1); (2,2);(3,3);(1,3);(3,1)}

now $(1,3)\in Rand(3,1)\in R\Rightarrow (1,1)\in R$

thus the relation is transitive also.

therefore we must add (3,1) to make it an equivalence relation.

hope this helps you

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