(1) Let be a function and . (i) Prove that . Prove that the converse is not universally true. Give a simple condition on which is equivalent to the converse. (ii) Prove that . Prove that equality is not universally true. (iii) Prove that .
(i) So . Then . Thus . The converse is . Suppose that and . Let . Then but . Is this correct? What would be the condition on which is equivalent to its converse?
(ii) which is a subset of . What would be a counterexample to show inequality?
(iii) This is basically the same as part (ii) except with an ?