Question

Which of the following are binary operation.justify your answer.
(i) the operation . defined on Q by a.b =a(b-a)

(ii) the operation . defined on [0, pi] by x.y = cosxy
also,for those operation which are binary operations,check whether they are associated and commutative .

Answer 1

clearly (i) is binary op, since rationals are closed under multiplication and addition.

(ii) is not, since x.y might produce a value not in [0,pi]. For any value xy in[pi/2,pi], cos(xy) < 0, so is not in the domain.

I think you can answer the other questions:

is a.(b+c) = a.b + a.c?
is a.b = b.a ?