QUIZ 3, Applied Mathematics, Senior Seminar in Mathematics

1) Cardinality is a key component of infinite sets. The set of natural numbers is countable. One of the cardinalities greater than countable is called uncountable. From the five options listed below, select the sets that are uncountable. There are 3 correct solutions out of 5

A. The real numbers in the interval

B. The irrational numbers in the interval

C. The real numbers in the interval

D. The integers in the interval

E. The rational numbers in the interval

2) Sequences are a key component in analysis. Understanding and categorizing them is critical in Applied Mathematics. Consider the sequence:

Select the terms that accurately describe this sequence. There are 3 correct answers out of 5.

A. Divergent

B. Infinite

C. Unbounded

D. Cauchy

E. Monotone

3) Use induction to prove the following equality:

4) The approach of mathematics has changed over the years. Describe how the approach of mathematics began starting in Ancient Babylonia, and then discuss how it changed during the time of the Ancient Greeks. Then, compare this to how mathematics is approached today. Be specific in your description and explanation.

5) The ability to travel efficiently around a graph has several practical applications. Eulerian and Hamiltonian graphs play an important part in many applications. The following graph, G, is Hamiltonian but NOT Eulerian. Analyze the 5 statements shown below and select the 3 that are true.

D

C

A

B

E

A. G is Eulerian after the edge AB is removed

B. G is Hamiltonian after the edge BD is removed

C. G is Hamiltonian after the edge AD is removed

D. G is Eulerian after the edge AD is removed

E. G is Hamiltonian after the edge AB is removed

6) Graphs appear in a wide variety of forms. Below is the adjacency matrix for a graph G:

Select all of the terms which accurately describe G. There are 3 correct answers out of 5.

A. Bipartite

B. Connected

C. Planar

D. Tree

E. Simple

7) Applied Mathematics is often concerned with transforming a difficult problem into a tractable framework. Homeomorphisms can be used to map a problem from one topological space to another while preserving all of the key attributes of the problem. Consider the real numbers with the standard topological basis, , of open intervals:

Also, consider the subset with the subspace topology.

Define a homeomorphism, , and prove that and are topologically equivalent.

8) The idea of a basis is fundamental in several areas of Applied Mathematics. This question addresses bases from a topological perspective. Consider the set of real numbers, . Which of the following sets is a basis for a topology on ? There are 3 correct answers out of 5.

A.

B.

C.

D.

E.

9) In a class 40% of the students enrolled for Math and 70% enrolled for Economics. If 15% of the students enrolled for both Math and Economics, what % of the students of the class did not enroll for either of the two subjects?

A. 5%

B. 15%

C. 0%

D. 25%

E. None of these

10) Answer all three parts.

(a) Find the Fourier coefficients corresponding to the function

??(??)={?? ???