give an example of an associative system ( S,*) with identity for which cancellation laws fail. Solution The composition on B^B satisfies neither right nor left cancellation laws: for example: a) f : R R f(x) = x^2 g : R R g(x) = x g = idR h : R R h(x) = x then f g(x) = x^2 , f h(x) = (x)^2 = x^2. So f g = f h, but g does not equal to h.