Let R be a binary relation on N 2 defined by: (a, b) R (x, y) if and only if a+y

Question

Let R be a binary relation on N2 defined by: (a, b) R (x, y) if and only if a+y = b+x.

Prove that R is an equivalence relation.

Explanation / Answer

1. Reflexive: prove that (a,b)R(a,b) that is to show that a+b=b+a, which is always true. So it is reflexive. 2. Symmetric: we have to prove that (a,b)R(x,y)= (x,y)R(a,b) we will take first (a,b)R(x,y) => a+y=b+x =>b+x=a+y =>x+b=y+a =>(x,y)R(a,b) Hence symmetric 3. Transitive: We have to show that if (a,b)R(x,y)is true and (x,y)R(p,q) is true, then (a,b)R(p,q) is also true. if (a,b)R(x,y)is true => a+y=b+x => x=a+y-b if (x,y)R(p,q)is true => x+q=y+p =>x=y+p-q equating both we get a+y-b=y+p-q cancelling 'y' on both side we get a-b=p-q =>a+q=b+p =>(a,b)R(p,q) So it is transitive. 4. Equivalence: As the relation is reflexive, symmetric and transitive, hence it is proved that the relation is Equivalence relation.

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