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

If S is the set of all rational numbers except 1 and * be defined on S by a**b =a+b-ab , for all a, b in S . Prove that (i) * is a binary operation on S. (ii) * is commutative as well as associative.

Consider the binary operation **:R xxR to R and o:R xx R to R defined as a**b = |a-b| and a o b = a . For all a , b in R . Show that * is commutative but not associative, .o. is associative but not commutative.

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.

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 a binary operation.

(i) Is the binary operation defined on set R, given by a* b=(a+b)/(2) AA a,b in R commutative? (ii) Is the above binary operation associative?

On the set R of real numbers, a binary operation o is defined by a0b=a+b+a^2b AA a, b inR . Prove that the operation is neither commutative nor associative.

Find the number of binary operations on the set {a, b}.

Find the number of binary operations on the set {a,b}.

For the binary operation a ** b = 3a + 2b for a, b in Z test whether it is associative.