MATH 409.501
Examination 1
February 15, 2012
Name:
ID#:
The exam consists of 3 questions. The point value for a question is written next to the
question number. There is a total of 100 points. No aids are permitted.
1. [40] In each of the following eight cases, indicate whether the given statement is true
or false. No justification is necessary.
(a) Let E be a set and suppose that there exists a surjective function f : R → E. Then
E is uncountable.
(b) If E is a subset of R which has a supremum, then the set −E = {−x : x ∈ E} has
an infimum.
(c) Let a ∈ R. Then |a| < ε for all ε > 0 if and only if a = 0.
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(d) If {Ex }x∈R is a collection of finite sets indexed by the real numbers, then
is at most countable.
S
x∈R
Ex
(e) Every subset of R has at most two suprema.
(f) Let f : R → R be defined by f (x) = x2 . Then f −1 ([0, 1]) = [−1, 1].
(g) Let A1 , A2 , A3 , . . . be nonempty finite subsets of N such that An ∩ Am = ∅ for all
distinct n, m ∈ N. Define the function f : N → N by declaring f (n) to be the least
element of An . Then f is injective.
(h) Let A1 , A2 , A3 , . . . be nonempty bounded subsets of R such that An ∩ Am = ∅ for
all distinct n, m ∈ N. Define the function f : N → R by f (n) = sup An . Then f is
injective.
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2. [30] (a) State the well-ordering principle.
(b) Prove that 2n−1 ≤ n! for all n ∈ N.
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3. [30] (a) State the completeness axiom for R.
(b) Let A be a nonempty bounded subset of R, and consider the set B = {x2 : x ∈ A}.
Prove that sup B exists.
(c) Give an example to show that the equality sup B = (sup A)2 may fail in part (b).
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