Math 414: Analysis I
Practice Exam 1
Spring 2014
Name:
*Please be aware that this practice test is longer than the test you will see on February
17, 2014. Also, this test does not cover every possible topic that you are responsible for
on the exam. For a comprehensive list of all topics covered on the exam, please see the
exam topics document on the website.
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) Define the following sets: the direct image f (A), the inverse image f −1 (C)
and what it means for a function f to be injective.
(b) Suppose that f : X → R is an injective function. Prove that f (A ∩ B) =
f (A) ∩ f (B).
(c) Construct an example to show that if f is not injective, then (b) need not
hold.
2. (a) Let A := 1 − n1 : n ∈ N . Determine sup A and/or inf A if they exist.
(b) Determine an M such that |f (x)| ≤ M if f : X → R, with f (x) = x2 − 6x − 2
and X = {x ∈ R : −2 ≤ x ≤ 1}. Hint: Use the generalized triangle inequality.
∞
(c) Let An := {(n + 1) k : k ∈ N}. Find ∪∞
i=1 An and ∩i=1 An .
3. Let A = B = {x ∈ R : −1 ≤ x ≤ 1} and consider the subset C := {(x, y) : x2 + y 2 = 1}
of A × B. Is this set a function? Explain.
4. Determine if the following statements are true or false. If the statement is true,
prove it. If the statement is false, provide a counterexample.
(a) Suppose that A, B ⊂ R. If A ⊂ B, then sup(A) ≤ sup(B).
(b) Suppose that A and B are sets with the property that sup(A) ≤ sup(B). Then
there is an element b ∈ B that is an upper bound for A.
(c) Suppose f : X → R and g : X → R are functions with nonempty domains.
If the ranges R(f ) and R(g) are bounded from above, then sup(f + g) ≤
sup f + sup g. (Here, sup(f ) = sup(R(f ))).
(d) The set of irrational numbers I = R \ Q has a completeness property.
5. Let A ⊂ R and B ⊂ R be two bounded subsets of R. Define
C = {x − y : x ∈ A, y ∈ B}
Prove that C is bounded in R and represent sup C in terms of bounds of A and B.
6. Suppose that y ∈ R, and y > 0. Prove that there exist an n ∈ N such that
n − 1 ≤ y < n.
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Math 414: Analysis I
Practice Exam 1
Spring 2014
7. (a) State the triangle inequality for any x, y ∈ R.
(b) State and prove the generalized triangle inequality.
8. (a) Let S ⊂ R be nonempty. Show that if u := sup(S), then for every number
n ∈ N the number u − n1 is not an upper bound for S, but the number u + n1
is an upper bound of S.
(b) Prove that the converse of (a) is true.
9. Prove that in an ordered field, if y < z, and x > 0, then xy < xz.
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