Let `**` be a binary opertion on the set Q of rational numbers as follows: ` a ** b = (a-b) ^(2)` commutative and associative.

Number of binary opertions on the set {a,b} are

Let ** be a binary operation on NN , the set of natural numbers defined by a**b=a^(b) for all a,binNN is ** associative or commutative on NN ?

If ** be the binary operation on the set ZZ of all integers, defined by a**b=a+3b^(2) , find 2**4.

Let A = N xx N and ** be the binary opertion on A defined by (a,b) ** (c,d) = (a + c, b+d) Show that ** is commutative and associative.

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 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.

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 operation on QQ_(0) (Set of all non-zero rational numbers) defined by a**b=(ab)/(4),a,binQQ_(0) . The identity element in QQ_(0) is e , then the value of e is--

Let ** be the binary opertion on N defined by a **b=H.C.F. of a and b. Is ** commutative ? Is ** associative ? Does there exist identity for this binary opertion on N ?

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 .