Let A and B be non-empty sets and let f:A→B be an arbitrary function.
a. (i) State the definition of the identity function on a set A.
b. (ii) Prove that f∘IA=f and name the set on which both sides are defined.
c. (iii) Prove that IB∘f=f and name the set on which both sides are defined.
d. (iv) Explain briefly why the equalities in parts (ii) and (iii) do not contradict the fact that, in general, function composition is not commutative.