portfolio questions

8/9/2019 Portfolio Questions

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MATH 211: Portfolio Assignment

The purpose of the portfolio is to provide a capstone piece which serves as evidenceof your thoughtful reflections and showcases what you have learned in this course.

Your portfolio will include:

1. a cover letter that stands as a reflection on why you chose the problems youincluded and how they demonstrate your strengths and your improvement in

the course2. the Portfolio Checklist which lists the problems that you have chosen

3. 10 problems (6 must be proof problems) chosen from the list provided on thissheet

The final portfolio will be collected at the end of the semester with drafts duetwice during the semester. Late work will be penalized one letter grade per day.

Unlike our normal homework assignments, you should be working on theseproblems independently. You should not share your ideas or progress with yourclassmates. In fact, you should not even tell each other which problems you are work-

ing on. Do not use the internet.

Format

The cover letter must be approximately one page, typed, single spaced, TimesNew Roman (or similar) 12-point font, and addressed to me, Professor Koss.

Each of the 10 problems must be written in LaTeX (using the Portfolio Templateon Moodle) and submitted in paper form.

First Draft Due: Thursday, March 27

4 problems (2 short answer and 2 proofs) written in LaTeX. You will turn in

one copy, and you will receive feedback from Professor Koss.

List of Problems Due: Thursday, April 10

Hand in the Portfolio Checklist with final list of problems (see Portfolio Check-list for specific directions).

Second Draft Due: Thursday, April 17

10 problems (4 short answer and 6 proofs) written in LaTeX. You should bring3 hard copies to class and treat this draft as your final draft. You will receivefeedback from your group.

Final Portfolio Due: Thursday, April 24Hand in (as described above):

cover letter the Portfolio Checklist final versions of your best 10 problems written in LaTeX (4 short answer and 6

proofs) in paper form


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List of Questions:

Section A: These are short answer questions. You should give detailed explanations,but not formal proofs.

1. You meet three inhabitants of Smullyans Island, and have the following con-versation. Who, if anyone, is telling the truth?

Asays, If B is lying, then so is C

B says, If C is lying, then so is A Csays, If A is lying, then so is B

2. Trooper Jones walks into Goldilocks Pub and sees four Boatsville College stu-dents (Al, Betty, Cindy, and Dan) enjoying various beverages. She asks thebartender, Is anyone here breaking the drinking law? The bartender replies,Everyone here is obeying the law.

In front of each person, there is a card which has the persons age on one sideand what he or she is drinking on the other side. Trooper Jones sees that theface-up sides of the cards look like Figure 1.

The drinking age law states in effect, If you are drinking beer, then you are atleast 21 years of age.

(a) Identify a set D and predicates P(x) and Q(x) so that the bartenders

statement is of the form For all xD, ifP(x) then Q(x).(b) What cards does Trooper Jones need to turn over to check that everyone

is obeying the law? Explain your answer.

3. Here is a Lewis Carroll puzzle. What conclusion can you draw from the state-ments?

Animals are always mortally offended if I fail to notice them; The only animals that belong to me are in that field;

No animal can guess a conundrum, unless it has been properly trained in

a Board-School; None of the animals in that field are badgers; When an animal is mortally offended, it always rushes about wildly and

howls;

I never notice any animal, unless it belongs to me; No animal, that has been properly trained in a Board-School, ever rushes

about wildly and howls.


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4. Draw a Venn diagram for four setsA, B, C, and D. Be sure to have a regionfor each of the sixteen possible sets such as A B CcD. Note: this problemcannot be done using just circles, but it can be done using rectangles.

5. Find a logically equivalent statement top that contains the statements p and qand symbols from the list,,.

6. In class, we introduced the inclusive or and the exclusive or by using the

examples of a waiter saying milk or sugar and coffee or tea. Give two othersentences in the English language that illustrate the inclusive or and twosentences that illustrate the exclusive or.

7. On August 30, 2005, Wikipedias entry on Modus Tollens gave the followingexplanation:

An argument can be valid even though it has a false premise. Such an argument

usually reaches a false conclusion. For example:

If an argument is modus tollens and both its premises are true, then it is sound.

One or both premises are false.

Therefore, the argument is unsound.Source: Modus Tollens, Wikipedia (http://www.wikipedia.org/)

Comment on the form and validity of the example that appeared on Wilkipedia.

8. Using a book that you borrow from me, write a one to two paragraph summaryof one of the following topics: Correctness of Algorithms, the Halting Problem.

Section B: Some of these statements are true, and some are false. Prove the onesthat are true, and give a disproof by counterexample for those that are false.

1. For any real number x, ifx if irrational then

1

x is irrational.

2. Ifa is any even integer and b is any odd integer, then a2 + b2 is odd,

3. 3

2 is irrational.

4. The difference in the squares of any two consecutive integers is odd.

5. For all integersmand n, ifamod 7 = 5 and bmod 7 = 6 then abmod 7 = 2.

6. The sum of any three consecutive integers is divisible by 3.

7. IfA={

x

Z

|x= 6a + 4 for some integer a

}, and B =

{y

Z

|y = 18b

2 for

some integer b}, then A= B .8. Ifr is any rational number, then 3r2 2r+ 1 is rational9. The sum of two positive irrational numbers is irrational.

10. For all integersn4, n!> n2.11. For any integer r, r4 has the form 8mor 8m + 1 for some integer m.


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12. The quotient of any two rational numbers is rational.

13. For all integersn3,

43 + 44 + 45 + . . . + 4n =4(4n 16)

3 .

14. The difference between any two odd integers is even.

15. For all integersaand b, ifa|10bthen a|10 or a|b.16. For all integersn >2, 5n + 9 < 6n.

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