Show that the relation (R) defined by (a,b)R(c,d) -> a+d=b+c on the set N X N is an equivalence relation

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 For every element (a,b) belongs to NxN

(a,b)R(a,b) implies a+b=b+a          (which is true)

Hence R is reflexive

Let (a,b)R(c,d) implies a+d=b+c

                           "              b+c=a+d

                           "              c+b=d+a

                           "              (c,d)R(a,b)

Hence R is symmetric

Let (a,b)R(c,d) and (c,d)R(e,f)

implies a+d=b+c and c+f=d+e

"             a+d+c+f=b+c+d+e

"            a+f=b+e

"            (a,b)R(e,f)

Hence R is transitive

 

Hence R is an equivalence relation

                                   

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