A binary operation `**` on `QQ` the set of all rational numbers is defined as `a**b=(1)/(2)ab` for all `a,binQQ`. Prove that `**` is commutative as well as associative on `QQ`.

Let @ be a binary operation on QQ , the set of rational numbers, defined by a@b=(1)/(8)ab for all a,binQQ . Prove that @ is commutive as well as associative.

Let * be a binary operation on Q_0 (set of non-zero rational numbers) defined by a*b= (3a b)/5 for all a ,\ b in Q_0 . Show that * is commutative as well as associative. Also, find the identity element, if it exists.

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 .

A binary operation ** on QQ_(0) , the set of all non-zero rational numbers, is defined as a**b=(1)/(3)ab for all a,binQQ_(0) Prove that every element of QQ_(0) , is invertible and find the inverse of the element (3)/(5)inQQ_(0).

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.

If S is a set of all rational numbers except 1 and ** be defined on S by a ** b = a + b-ab for all a, b in s then prove that ** is commutative as well as associative.

For a binary operation ** on the set of rational numbers Q defined as a** b= (ab)/2 Determine whetler ** is commutative

For a binary operation ** on the set of rational numbers Q defined as a** b= (ab)/2 Determine whetler ** is associative

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^(+) .