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Final Exam: Math Camp

Summer 2022

1 Basics

Question 1.1 (25 points)

Define binary relations  and  on R 2 as follows. For any x, y ∈ R 2 ,

Please complete the following statements with the following qualifiers. You do not need to provide a proof or a counter-example.

A. both and             B. only

C. only                        D. neither nor

(a)         is (are) reflexive.

(b)         is (are) complete.

(c)         is (are) transitive.

(d)         is (are) antisymmetric.

(e) The range and the domain of         are the same set.

2 Analysis

Question 2.1 (25 points)

Please determine whether the following statements are true or false. You only need to write down “T” or “F”. You do not need to provide a proof or a counter-example.

        (a) The intersection of an open set and a closed set is neither open nor closed.

        (b) A = [0, 1]\Q is a closed set in R.

        (c) If A is closed and B is open, then A\B is closed.

        (d) Suppose f(x) : [a, b] → R is continuous. If for any x ∈ [a, b], there exists y ∈ [a, b] such that |f(y)| ≤ 1/2 |f(x)|, then there exists x0 ∈ [a, b] such that f(x0) = 0. (Hint: Try using the extreme value theorem.)

        (e) Suppose f(x) : R → R is continuous. If A ⊆ R is open, then, f(A) is open.

Question 2.2 (25 points)

Let X be a nonempty set and let d : X × X → R be such that ∀x, y ∈ X

(a) (8 points) Show that (X, d) is a metric space.

(b) (8 points) What is Bε(x) in (X, d)?

(c) (9 points) Let (xn) be a sequence in (X, d). Show that if xn → x, then {xn : n ∈ N+} is a finite set.

3 Static Optimization

Use the KKT conditions to find the closest (under the d2 metric) point to (0, 0) in the following set:

M = {(x, y) ∈ R 2 : x + y ≥ 4, 2x + y ≥ 5}.