1. (a) Define, with precision and in a form suitable for usingin a proof, the least upper bound of a nonempty subset S ? R thatis bounded above.
(b) Define, with precision and in a form suitable for using in aproof, an open set in a metric space (X, d).
(c) Give an example, if possible of a function f : X ? Y andsubsets A, B ? X such that f(A ? B) is not equal to f(A) ?f(B).
(d) Give an example, if possible, of a subset S ? R that isbounded above, but that has no least upper bound.
(e) Define f : R ? R 2 by f(t) = (t, t2 ) and let E = {(x1, x2): x1 < 1, x2 < 4}. Find f ?1
(E). (f) Is there a set A and an onto function f : A ? P(A)?
