Consider the binary operations `""^('ast')` and 'o' on R defined as `a^(ast)b=abs(a-b) and aob=a`, for all `a, b in R`. Show that `a^(ast)(boc)=(a^(ast)b)oc` for all a, b, c `in` R.

Consider the binary operations *: RxxR->R and o: RxxR->R defined as a*b=|a-b| and aob=a for all a ,\ b in Rdot Show that * is commutative but not associative, o is associative but not commutative.

Consider the binary operations ** \: R\ xx\ R -> R and o\ : R\ xx\ R -> R defined as a **\ b=|a-b| and aob=a for all a,\ b\ in R . Show that ** is commutative but not associative, o is associative but not commutative.

Consider the binary operation '*' on R-{-2} defined a * b =a+b+(ab)/(2) , for all a,b in R-{-2} . Find the value of x if 1 * (x * 2) = 8.

The binary operation R xx R to R is defined as a*b=2a+b . Find (2*3)*4 .

A binary operation ""^(ast) defined on N, is given by a^(ast)b= H.C.F. (a, b), foralla, b in N . Check the commutativity and associativity.

If the binary operation ** , defined on Q , is defined as a**b=2a+b-a b , for all a **\ b in Q, find the value of 3**4

For the binary operation * defined on R-{1} by the rule a * b=a+b+a b for all a ,\ b in R-{1} , the inverse of a is

Discuss the commutativity and associativity of the binary operation * on R defined by a*b=(a b)/4 for all a ,b in R dot

Consider the binary operation * : R xx R rarr R and o : R xx R rarr R defined as a * b = |a-b| and a o b = a, AA a, b in R . Is o distributive over * ? Justify your answer.

The binary operation defined on the set z of all integers as a ** b = |a-b| - 1 is