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.

A relation is an equivalence if and only if it is reflexive , symmetric and transitive.
R = {(1,1); (2,2); (3,3); (1,3)}
the given relation is reflexive as aRa for aR
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)R and (3,1)R(1,1)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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