1. Show that the relation R defined by R={(a,b):a-b is divisible by 3;a,b E N}is an equivalence relation .

Reflexivivity: For any , we have

 

Thus, for all . So, R is reflexive.

Symmetricity: Let

 

Thus,    for all . So, R is symmetric.

Transitivity:Let such that and . Then, 

 

So, R is transitive.

Thus, R is an equivalence relation.

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