Consider the binary operation:` R xx R rarr R ` defined as : `a** b = |a+b| a,b in R`. Show that * is commutative.

In the binary operation *: Q xx Q rarr Q is defined as : a**b = a +b + ab, a,b in Q Show that * is commutative.

In the binary operation *: Q xx Q rarr Q is defined as : a*b = a + b - ab, a,b in Q Show that * is commutative .

In the binary operation *: Q xx Q rarr Q is defined as : a**b=(ab)/4, a,b inQ Show that * is commutative.

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

In a binary operation **: phi xx phi rarr phi is defined as a ** b = (ab)/4, a , b in phi . Show that ** is associative.

Consider the binary operations * : R xx R 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.

Consider the binary operations * : RxxRrarrR and o : RxxRrarrR defined as a*b = |a – b| and aob = a, forall a, b in R Show that ∗ is commutative but not associative, o is associative but not commutative. Further, show that ∀ a, b, c in R , a * (b o c) = (a * b) o (a * c) . [If it is so, we say that the operation ∗ distributes over the operation o]. Does o distribute over ∗? Justify your answer.

Let A = N xx N and ∗ be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d) Show that ∗ is commutative and associative. Find the identity element for ∗ on A, if any.