For S 1 2 3 4 5 and the relation R 1 1 1 5 5 5 5 1 4 4 3 1

For S = {1, 2, 3, 4, 5} and the relation R = {(1, 1), (1, 5), (5, 5), (5, 1), (4, 4), (3, 1), (4, 3)}. What must be added to make R an equivalence relation?

Solution

(1,3) , (3,4) , (3,3) must be added to make it an equivalence relation . As to make an equivalence relation it should satisfy reflexive, symmetric and transitive ... so by adding these elements it will satisfy all three of them .

Reflexive (a,a)€ R

Symmetric : if (a,b)€R then (b,a) € R

Transitive : if (a,b)€R , (b,c)€R then (a,c)€ R