Math 414: Analysis I
Exam 1 (80 points)
Spring 2014
Name:
*No notes or electronic devices. You must show all your work to receive full credit.
When justifying your answers, use only those techniques that we learned in class.
1. Suppose that f : X → R is a function and suppose that A, B ⊂ X, and C, D ⊂ R(f )
where R(f ) denotes the range of f .
(a) (5) Define the following sets: the direct image f (A), and the inverse image
f −1 (C).
(b) (10) Recall that C \ D = {x : x ∈ C and x 6∈ D}. Prove that f −1 (C \ D) ⊆
f −1 (C) \ f −1 (D).
(c) (5) It turns out that if we replace the inverse image with the direct image,
then (b) no longer holds. To show that this is true consider the function f :
[−1, 1] → R with f (x) = 0 for all x ∈ [−1, 1]. Let A = [−1, 0] and let B = [0, 1].
Calculate f (A \ B) and f (A) \ f (B) to see that f (A \ B) 6⊆ f (A) \ f (B).
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Math 414: Analysis I
Exam 1 (80 points)
Spring 2014
2. (a) (5) Define what it means for a function f : A → B to be surjective and
injective.
(b) (10) Define the function f : N → N by
f (x) = x2 + 5.
Prove that f is an injective function.
(c) (5) Is f a bijective function? Justify your answer.
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Math 414: Analysis I
Exam 1 (80 points)
Spring 2014
3. Determine if the following statements are true or false. If the statement is true,
prove it. If the statement is false, provide a counterexample.
n
(a) (5) If A = n(−1) : n ∈ N then A has no upper bounds.
(b) (5) If b < L for every element b in the set B, then sup(B) < L.
(c) (5) If |c + d| = |c| + |d|, then c ≥ 0 and d ≥ 0.
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Math 414: Analysis I
Exam 1 (80 points)
Spring 2014
4. (25) Do either (a) or (b).
(a)
i. Suppose that A ⊂ R is a set that is bounded above, and B ⊂ A. Prove
that B is bounded above and that sup(B) ≤ sup(A).
ii. Consider the sets A and B as given in (i), and suppose that whenever
x ∈ A there is a y ∈ B such that x ≤ y. Prove that sup(B) = sup(A).
(b)
i. Prove that n < 2n for all n ∈ N, where N denotes the set of natural
numbers.
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ii. If y > 0, prove that there is a n ∈ N such that n < y.
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