(Z,*) where a* b = a+b-ab for all `a,b inZ` prove that the given binary operation * is associative and commutative.

An operation ** on ZZ , the set of integers, is defined as, a**b=a-b+ab for all a,binZZ . Prove that ** is a binary operation on ZZ which is neither commutative nor associative.

A binary operation ** on QQ , the set of rational numbers, is defined by a**b=(a-b)/(3) fo rall a,binQQ . Show that the binary opearation ** is neither commutative nor associative on QQ .

let ** be a binary on QQ , defined by a**b=(a-b)^(2) for all a,binQQ . Show that the binary operation ** on QQ is commutative but not associative.

let ** be a binary operation on RR , the set of real numbers, defined by a@b=sqrt(a^(2)+b^(2)) for all a,binRR . Prove that the binary operation @ is commutative as well as associative.

Let * be a binary operation defined by a* b = L.C.M , (a,b) AA a,b in N . Show that the binary operation * defined on N is commutative and associative. Also find its identity element of N.

Let P(A) be the power set of a non-empty set A and a binary operation @ on P(A) is defined by X@Y=XcupY for all YinP(A). Prove that, the binary operation @ is commutative as well as associative on P(A). Find the identity element w.r.t. binary operation @ on P(A). Also prove that Phi inP(A) is the only invertible element in P(A).

Show that +:R×R→R and o:R×R→R defined as a∗b=∣a−b∣ and aob=a for all a,b∈R. Show that ′ ∗ ′ is commutative but not associative, 'o' is associative but not commutative.

let ** be a binary operation on ZZ^(+) , the set of positive integers, defined by a**b=a^(b) for all a,binZZ^(+) . Prove that ** is neither commutative nor associative on ZZ^(+) .

Let S be a set of containing more than two elements and a binary operaton @ on S be defined by a@b=a for all a,binS . Prove that @ is associative but not commutative on .

Let ** be an operation defined on NN , the set of natural numbers, by a**b=L.C.M.(a,b) for all a,binNN . Prove that ** is a binary operation on NN .