Question

1. (a) Define, with precision and in a form suitable for using in a proof, the least upper bound of a nonempty subset S ⊂ R that is bounded above.

(b) Define, with precision and in a form suitable for using in a proof, an open set in a metric space (X, d).

(c) Give an example, if possible of a function f : X → Y and subsets 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 is bounded 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)?

Answer 1

(A)

(B)

(C)

(D)

(E)

(F)