Logic & Machines

Now..with corrections, which is quite contradictory to my theory of internet copying. We should be assessed on our original ideas, yet pressure to earn high grades persists...and so it is a competition to see who can log in most often to edit solutions and read other's solutions...I trust my professor's objectives, however, and am not concerned.

Note: ^ = AND
~ = NOT
...unfourtunately, I don't have a symbol for OR, so OR = OR.

2064 = 0(mod8), 2064=0(mod3), 2064=6(mod7)

(1) The 16 output functions, in order and hopefully with no careless errors, are:
1. (X ^ ~X) ^ (Y ^ ~Y)
2. X^Y
3. X ^ ~Y
4. X = (X^Y) OR (X ^ ~Y)
5. Y ^ ~X
6. Y = (Y^X) OR (Y ^ ~X)
7. (X ^ ~Y) OR (Y ^ ~X)
8. X OR Y
9. ~X ^ ~Y
10. (X^Y) OR (~X ^ ~Y)
11. ~Y = (X ^ ~Y) OR (~X ^ ~Y)
12. X OR ~Y
13. ~X = (Y ^ ~X) OR (~Y ^ ~X)
14. ~X OR Y
15. ~(X^Y)
16. (X^Y) OR ~(X^Y)

(2)a) All functions can be represented using {AND, NOT} because of De'Morgan's law...one day, when I have more time, I will show the originally suggested functions in terms of {AND, NOT}.

The following functions cannot be represented using only {AND, NOT}:
4. X
6. Y
7. XOR
8. OR
10. XAND
11. NOT Y
12. Y IMPL X
13. NOT X
14. NAND
16. 1

(3)a) The following functions are not commutative (and hence the other ones are commutative):
3. X ^ ~Y does not equal Y ^ ~X
4. X does not equal Y
5. Y ^ ~X does not equal X ^ ~Y
6. Y does not equal X
11. ~Y does not equal ~X
12. Y IMPL X does not equal X IMPL Y
13. ~X does not eqaul ~Y
14. X IMPL Y does not equal Y IMPL X

(4) There are 2^9 = 512 possible output functions when there are three possible inputs.

512 can also be found by the summation:
9C0 + 9C1 + 9C2 + 9C3 + 9C4 + 9C5 + 9C6 + 9C7 + 9C8 + 9C9 = 512

I discovered this method by examining the binary case:
6 ways to choose two 1s, 4 ways to choose one 1, 4 ways to choose three 1s, 1 way to choose no 1s, and 1 way to choose four 1s (or 0s in each case).

btw: I did think of 3^9 = 19683 possible output functions since in binary it is 2^4 = 16, but didn't like the logic behind that solution, so I discarded it.

(5) Three binary operations (AND, OR, and NOT), along with operations to ACCEPT AND/OR REJECT 0's, 1's, and 2's, are necessary to express every possible output function in trinary. XOR, IMPL, etc., are just compositions of these.
For example, XOR = (X ^ ~Y) OR (Y ^ ~X) in trinary as in binary. Where X:= ACCEPT 1 only. I am picturing trinary operating more as a Turing machine.
input output
00 0
01 1
02 0
10 1
11 0
12 1
20 0
21 1
22 0

(6)pic1 AND pic2





pic1 OR pic2



g) 0 = white, 1 = black

x/ y/ x-->y /x ^ (x-->y)
0/0/1/0
0/1/1/0
1/0/0/0
1/1/1/1

x ^ (x-->y) = x ^ y







(7)
P - poison
D - death
C - change in blood chemistry
R - residue of poison in stomach
B - puncture marks on the body
N - injection by needle

(P IMPL D) AND (C OR R). NOT C AND NOT R AND B. N IMPL B. (P IMPL D) OR NOT B.

Orders of Magnitude

(1) 63K = 2^6 x 2^10 bytes = 2^16 bytes; 1GB = 2^10 MB = 2^30 bytes. Today's computer has 2^14 bytes more memory.

(2) Floppy disk - 800 x 2^10 bytes = 100 x 2^13 bytes. DVD - 4.7 x 2^30 bytes. Dividing the DVD memory by the floppy disk memory...1 DVD = approximately 6160 Floppy disks.

(3) early 1990s: Apple - 110MHz, Intel - 100MHz (http://en.wikipedia.org/wiki/Clock_rate). Today, computer can run at 500GHz, Intel just sells a 3.40 GHz Processor (http://processorfinder.intel.com/details.aspx?sSpec=SL7PP).

3.4 x 2^10 MHz v.s. 110 MHz. Today's computer is about 35x faster?

See the link below to more acurately compare speed factors, contains some complicated details about clock rate, bits/clock, and multiplier, to get computer "speed". http://www.yale.edu/pclt/PCHW/clockidea.htm

A Tool to Improve Assessment in Math Classrooms in light of vast Computer Applications

A computer program that will take an electronically generated math assignment and create clones with different numbers would be nice. I create different exam versions to prevent cheating, but if there were a computer program that would make this process more efficient (than say Excel), essentially 30 unique assignments could be generated so that students could work together to understand solutions to problems, but they would be responsible for doing all of the steps themselves to get the answers. Working out solutions together is not copying. It is not cheating if the students must use some independent thought to prepare solutions to assignment problems. (Hopefully these last two lines are not contradictory, and that my point is clear.) In addition, just "changing the numbers" avoids unfair assessment practices that assignments including entirely different questions are at risk of.

Copying from the Internet is not cheating?

I remain convinced that the Internet facilitates copying primarily in subjects other than Math & Science. Opinions about the role of homework in the math classroom can only be formed after one has clarified the definition of cheating as it pertains to their course.

The following web site provides links (a particularly good one titled "Electronic School: Digital Deception") supporting the growing concern of cheating in the classroom: http://www.gradebook.org/Cheating.htm.

Common expectations are that students should hand in work that includes only their own ideas/efforts (i.e. not copy homework solutions irregardless of the source), and not have access to electronic memory during closed-book assessments.

Technology makes it easier to cheat during a test, but this is why there are testing guidelines such as no cell phones, ipods, etc. during a test. I do not think that this needs to be of concern, provided the teacher is on top of things in their classroom (and able to handle the supervision of the number of students in their class).

As for copying assignment solutions? Teachers must develop strategies to deal with this reality because there is no way of preventing it. It is still far easier for students to copy one anothers solutions than to plagerize a web solution. Actually, I think that the process of searching for a math problem on the Internet (or in a book) is a beneficial activity. The student is still thinking about how to formulate a solution. Perhaps there are online examples, with different numbers, that will provide enough support for the student to be able to answer the question? I prefer this from a student to a complete lack of motivation to complete an assignment.

Students were able to share information about assignments with each long before the Internet. How technology, computers in particular, does facilitate cheating is that it makes the process easier and quicker.

That being said, teachers need to develop strategies like designing different methods of assesment that discourage/prevent copying, knowing their students abilities in order to recognize a copied assignment, and creating a classroom environment in which students will want to turn in their own work. See some great applications of this (along with more research on computers & cheating) at http://www.educationworld.com/a_admin/admin/admin375.shtml/.