Assignment:
Let F be the set of all functions f : R → R. We know that is an abelian group under the usual function addition, (f + g)(x) = f(x) + g(x). We define multiplication on F by (fg)(x) = f(x)g(x). That is, fg is the function whose value at x is f(x)g(x). Show that the multiplication defined on the set F satisfies axioms R2 and R3 for a ring.
R2: Multiplication is associative.
R3: For all a, b, c ? R, the left distributive law, a.(b+c) = (ab) +(ac) and the right distributive law (a+b)c = (ac) + (bc) hold.
*Note that R is the ring.
Provide complete and step by step solution for the question and show calculations and use formulas.