Q. 204.7( 3 Votes )

Let N denote the

Answer :

Given that, R be the relation in N ×N defined by (a, b) R (c, d) if ad(b + c) = bc(a + d) for (a, b), (c, d) in N ×N.


Reflexivity:


Let (a,b) R (a,b)


ab(a+b) =ba( a+b)


which is true since multiplication is commutative on N.


R is reflexive.


Symmetricity:


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


ad(b + c) = bc(a + d)


da(c + b) = cb(d + a)


[since addition and multiplication is commutative on N]


cb(d + a) = da(c + b)


(c,d) R (a,b)


(a,b) R (c,d) (c,d) R (a,b) (a,b), (c,d), (e,f) N ×N.


R is symmetric.


Transitivity:


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


ad(b + c) = bc(a + d)




also (c,d) R (e,f)


and cf(d + e) = de(c + f)




Adding (i) and (ii) , we get





af(b + e) = be(a + f)


(a,b) R (e,f)


Thus , (a,b) R (c,d) and (c,d) R (e,f) (a,b) R (e,f) (a,b), (c,d), (e,f) N ×N.


R is transitive.


Hence, R is an equivalence relation.


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