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 .

Show that the operation ** on ZZ , the set of integers, defined by. a**b=a+b-2 for all a,b inZZ (i) is a binary operation: (ii) satisfies commutaitve and associative laws: (iii) Find the identity elemetn in ZZ , (iv) Also find the inverse of an element ainZZ.

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 .

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 ** 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.

Show that an operation ** on RR , the set of real numbers, defined by a**b=3ab+sqrt2, for all a,binRR . Is a binary operaion on RR.

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.

Prove that the operation ** on ZZ defined by a**b=a|b| for all a,binZZ is a binary operation

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 .

Show that the binary operation ** defined on RR by a**b=ab+2 is commutative but not associative.