Show that `**: R × R to R` defined by `a **b=a + 2b` is not commutative.

Show that **: R xxR to R given by a ** b to a + 2b is not associative.

On Q defined * by a ** b = ab + 1 show that * commutative.

Let * be a binary operation on the set R defined by a ** b = (a + b)/2 . Show that * is commulative but not associative.

Show that +: R xx R to R and × : R xx R to R are commutative binary operations , but - : R xx R to R and div : R _(**) xx R _(**) are not commutative.

Show that **: R xx R to R given by (a,b) to a + 4b ^(2) is a binary operation.

Show that the binary operation * defined on R by a ** b = (a + b)/2 is not associative by giving a counter example.

Let * be a binary operation on N defined by a ** b = HCF of a and b. Show that * is both commutative and associative.

Show that the function f : R to R defined by f(x) = x^(2) AA x in R is neither injective nor subjective.

A binary operation * on N defined as a ** b = a^(3) + b^(3) , show that * is commutative but not associative.

Let * be a Binary operation defined on N by a ** b = 2^(ab) . Prove that * is commutative.