From mpf@cise.ufl.edu Mon Sep 20 19:57:19 1999 Date: Mon, 20 Sep 1999 19:57:19 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT 3100 students) Subject: Homework #3 hints X-UIDL: e112cf26ec80c26ff447aa0219cf21a5 Here are some hints on the parts of homework #3 that deal with book sections 1.6 and 1.7. Hints for 1.8 will come later. -Mike Section 1.6 ----------- Ex. 4: For part d, you might want to review the exact definition of bit strings, which was covered in an earlier section. (To find it, it's good exercise to try using the book's index.) Ex. 14: After you've reviewed the definition of a bijection, it may be helpful to draw the graphs of these functions. A function f(x) is one-to-one if the equation f(x)=y has at *most* one solution for x, given any value of y. The function is onto if there is at *least* one solution for any given value of y. Imagine drawing a value for y (a horizontal line) across the function's graph at various heights, and think about how many times that line might intersect the curve of the given function. You may also find section 1.6, definition 6 (p. 59) (which discusses strictly increasing functions), together with the text that immediately follows it on p. 60, to be useful. (In general, please be sure to always read through the whole text section before starting the exercises, as not every detail can be covered in lecture.) Ex. 28: This problem is asking you to compute the inverse images of some sets, under the function f(x)=xx (x squared). Please carefully read the definition of "inverse image," given just before the problem statement, and try to understand it. Basically, the inverse image of a set S (under a function f) is just the set of all values in the domain whose images (under f) are in S. Please give your answers in the simplest form that you can. Ex. 52: When drawing function graphs, it is conventional to use open circles to indicate points that are absent from the graph, but that are limit points that are approached arbitrarily closely by the graph. These are useful for drawing functions that involve floor/ceiling, since these functions generally contain discontinuities at the points where a floor or ceiling value clicks over from one integer to the next. Section 1.7: ------------ Ex. 18: You do *not* have to try to tranform these summations to some closed form before evaluating them. Ex. 20: I said in class that in general you don't have to aggressively simplify closed form expressions, but in this case, it's easy to put the answer in a very simple form. Ex. 28: If you have a summation (or a product) with an index i that ranges from a lower bound m (written below the summation or product symbol) to an upper bound n, this means summing (or multiplying) over (integer) values of i such that m <= i <= n is true (where "<=" means less than or equal to). Therefore, if it happens that m>n, then you are summing (or multiplying) over an *empty* set of values (as opposed to summing (or multiplying) *backwards* over the range from n up to m). A summation over an empty set of index values is considered by definition to be 0 (the additive identity). Similarly, a product over an empty set of index values is considered to be 1 (the multiplicative identity). You can use these facts in your answer if you wish. Ex. 38: This is an A-level question, so I don't expect all students necessarily to get it. To solve this, you will need to come up with a bijective function between the rationals between 0 and 1, and the natural numbers. One way to do this is to describe a method for listing (as in a sequence or series) the rationals between 0 and 1 in a specific order, such that every rational number will be eventually covered in your list, but never more than once. Then, talk about the index function of this series. You don't have to try to define the terms of the series using a closed-form algebraic expression. Rather, just try to explain, in clear, mathematical English, a methodical procedure for generating the terms of the series, in the order you have chosen. (The hint given in the book is a useful starting point, but is not detailed enough.) An alternative, somewhat different approach to the whole problem uses the result of exercise 37. [An interesting though side point relating to this problem is that if *all* of a set's members have finite, discrete descriptions (identifiers) of some sort, then it follows that the set is countable. Or equivalently, *any* uncountable set must contain some members that have *no* finite description. So, for example, the set of real numbers can be uncountable only because some of the real numbers actually require an *infinite* amount of information to specify. Any rational number, on the other hand, has a finite description. (You aren't allowed to use this argument to solve this problem though.)] From mpf@cise.ufl.edu Mon Sep 20 16:06:47 1999 Date: Mon, 20 Sep 1999 16:06:47 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT3100 students) Cc: pau@cise.ufl.edu, wvalella@cise.ufl.edu, xgu@cise.ufl.edu, eisland@ecel.ufl.edu Subject: COT3100 quiz 2 solutions and notes. X-UIDL: 0f0174a77c50f0d3e0c25310139274ce COT3100 students, Quiz #2 is now posted online for your reference, at http://www.cise.ufl.edu/~mpf/quiz2.html. I have also written up annotated solutions to quiz 2, enclosed below and also posted in a text file at http://www.cise.ufl.edu/~mpf/quiz2-notes.txt. Due to a rather low average score on quiz 2, I am reviewing the grading on it, to ensure that the TA's scoring coincides with what I had in mind. This will hopefully be finished Wednesday, and I will return quiz 2 at that time. Please be reassured that each student's lowest quiz grade of the semester will be dropped, and that I will strive to make future quizzes shorter and clearer. I will also be emailing some clarifications/hints on homework 3 shortly. If you have questions, please also try to come to office hours; again, the schedule is at: http://www.cise.ufl.edu/~mpf/ohs.html Best regards, Your professor, -Mike Frank ------------------------------------------------ Quiz 2 solutions and notes. In the following, we use the words NOT, FORALL, EXISTS, INTERSECT, and UNION in place of the corresponding predicate logic and set theory symbols, due to the limitations of the ASCII character set. Problem 1. Line 3: The previous line was NOT FORALL x NOT ( NOT P(x) \/ Q(x) ). The comment in brackets says to use one of DeMorgan's laws. One could use either of the following versions of one of his laws: NOT (p \/ q) <=> NOT p /\ NOT q p \/ q <=> NOT (NOT p /\ NOT q) Using the first version is easier here because it gives us fewer double negations to deal with. In any case, with a=NOT P(x), and b=Q(x), applying the law gives us one of the following new expressions, inside the NOT FORALL x: NOT NOT P(x) /\ NOT Q(x) NOT NOT (NOT NOT P(x) /\ NOT Q(x)) in either case, the double-negations can be immediately removed, giving just P(x) /\ NOT Q(X) after the NOT FORALL. Therefore we have just NOT FORALL x P(x) /\ NOT Q(x) in the blank on line 3. (You could leave in some of the double negations, but they have to come out sometime before line 5.) Line 4: The previous line was NOT FORALL x P(x) /\ NOT Q(x) and the comment in brackets tells you what to do next. Just split the forall into two parts. That is, the sub-expression FORALL x P(x) /\ NOT Q(x) becomes (FORALL x P(x)) /\ (FORALL x NOT Q(x)). It is very important to leave in the first set of parentheses, as these delimit the scope of that first FORALL. The second set are redundant, however. It is also very important to leave in the initial NOT, which applies to that *entire* subexpression corresponding to the initial FORALL. This must be indicated with a new *additional* set of parentheses, or else the NOT will only apply to the first FORALL, and the meaning will be wrong! NOT ((FORALL x P(x)) /\ (FORALL x NOT Q(x))) Line 5: This line just applies one of DeMorgan's laws, namely the one that takes NOT (a /\ b) to NOT a \/ NOT b, to the result of the previous line. If you wrote anything with "DeMorgan", or if you wrote the rule NOT (P /\ Q) <=> NOT P \/ NOT Q, or an equivalent, it was accepted. Line 6: The previous line was NOT (FORALL x P(x)) \/ NOT (FORALL x NOT Q(x)). The comment says to use the rule in which a FORALL expression is replaced by an EXISTS expression. Looking ahead to line 7, we can see that we end up with an expression that has a FORALL around the P(x) part, and an EXISTS around the Q(x) part. So in line 6, we will be changing the second subexpression, the one that contains Q(x): NOT (FORALL x NOT Q(x)). Applying the rule to this part, we get: EXISTS x Q(x) and plugging this back in to the formula gives NOT (FORALL x P(x)) \/ EXISTS x Q(x). As an additional check, we can see from line 7 that this can be directly transformed into the final answer, (FORALL x P(x)) -> EXISTS x Q(x) Problem 2. Some students were confused by the meaning of the parenthetical remark, "You can assume that A and B and U are chosen such that all the values in each list are different from each other." First, having some remark like this was necessary because if it was not included at all, then the order of the numeric quantities (values) in the two lists "a)" and "b)" would not be well-defined, because some of them might be the same. For example, if A and B were disjoint, then we would have |A INTERSECT B| = |{}| |A XOR B| = |A UNION B| = |A| + |B|. If A=B, we would have |A UNION B| = |A INTERSECT B| = |A| |A - B| = |{}|. If A=U, we would have |A| = |U|. The single remark "All the values in each list are different from each other" suffices to rule out all three of the above cases, which would otherwise have to be ruled out by three separate restrictions. Some people thought that by "each list," I meant "each set" (referring to sets A and B), and that by "values," I was referring to the "elements" or "members" of those sets. However, I will not consider this to be a valid interpretation of the question, for the following reasons: 1. "List" is not another word for "set". "Set" is the very specific type of discrete structure that we have been studying, and we never introduced "list" as an interchangable term for "set". 2. I used the word "list" earlier in the paragraph when I said "the following lists of quantities", referring to list a) and list b). I intentionally used the word "list" again in the parenthetical remark in order to make the connection with the earlier sentence. 3. If the remark had meant "All the elements in sets A and B are different from each other," that would have been partly redundant, because within any single set (A or B), all of the elements are already different from each other, by the definition of a set as containing no repetitions. 4. If I had meant that A and B had no elements in common, I would have just said that, or "You can assume that sets A and B are disjoint." 5. If A and B were disjoint, it would have been impossible to write all the given quantities in order of increasing size, since many of then would have actually been equal. So the question would have been impossible to answer. This should have been a clue that you were misinterpreting the problem. 6. From the grading, it seems that the great majority of students did interpret the problem correctly. Also, some people were confused by the expression |A|+|B|, thinking that "+" was some set operation not previously defined. However, the arguments to that expression were the cardinalities, |A| and |B|. The problem statement specified that A and B are finite sets, and you should know that the cardinality of any finite set is just an ordinary integer. Therefore, |A|+|B| just denotes the sum of two numbers. The (one and only) correct solution to problem 2 was: a) |{}|, |A INTERSECT B|, |A|, |A UNION B|, |U|. b) |{}|, |A - B|, |A XOR B|, |A UNION B|, |A|+|B|. although partial credit was given for partially correct answers. From mpf@cise.ufl.edu Tue Sep 14 12:48:50 1999 Date: Tue, 14 Sep 1999 12:48:50 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT 3100/7094X students) Subject: my-staff X-UIDL: 26e7dbab5c026e931fd76210e2aa9350 Hello class, Since classes tomorrow are canceled due to the hurricane, and we still need to cover sequences, summations, and orders of growth before you can finish the next assignment, the due date for Homework #3 is being pushed back to next Wednesday (Sep. 22). Quiz #3 will be Friday, Sep. 24. I do advise, however, that you go ahead and work on the exercises from section 1.6, which we have essentially finished. Good luck, and be careful not to get sucked into any tornadoes... :) Thanks, -Mike From mpf@cise.ufl.edu Fri Sep 10 15:44:02 1999 Date: Fri, 10 Sep 1999 15:44:02 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" Cc: pau@cise.ufl.edu, wvalella@cise.ufl.edu, xgu@cise.ufl.edu, eisland@ecel.ufl.edu Subject: General homework guidelines and tips X-UIDL: da77bfe4aca615951c21ae06466ed6a5 Hello, COT3100/7094X students: Here is some advice from the teaching assistants that might help if you are finding that more points than you would like are being deducted on your homework assignments. General Home Work Guidelines and Tips: 1. It is always better to show more work than to leave steps out. We have to see the progression of thought in an organized manner. If you don't understand what you have written, more than likely we won't either. 2. Even if the question has a simple yes/no or true/false answer be sure to explain your reasons. Just because the author hasn't explicitly asked you to show work, any mathematical answer should be supported through a logical flow of ideas. 3. Partial credit will be given even if you have the wrong answer, if you show us that you have understood the basic concepts (by following 1 and 2 above). 4. If you are not sure what the question is asking for, it is your responsibility to come to us for clarifications during office hours. 5. Many of the concepts needed to solve a problem can be found or clarified by reading and/or answering the problems directly preceeding it. 6. The more problems you do the better prepared you are, because concepts become more concrete. 7. Start the homework early, so that if you do have questions you do not have to wait in line. Make through you go through your solutions several times before you turn in the homework, because sometimes better answers present themselves with more thought. I might add that if you do find an actual error in the grading - for example, if the grader mis-read what you wrote, or took off an extra point due to an accidental slip of the pen, or summed the grade wrong - please do let us know, and we will fix it. But, please also be aware that if the grader did read your answer correctly and not slip up, from there on, the actual grading policy is determined by the TAs, as outlined above. Their policy decisions as to how many points to routinely award or deduct for various types of answers, omissions, and/or mistakes will be considered final. Good luck! -Mike From mpf@cise.ufl.edu Wed Sep 8 18:34:32 1999 Date: Wed, 8 Sep 1999 18:34:32 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT 3100/7094X students) Cc: ugrankar@ufl.edu, wvalella@cise.ufl.edu, xgu@cise.ufl.edu, eisland@ecel.ufl.edu Subject: 1.3 problem 18e X-UIDL: ede003ab2ee3fe392f5d6c62ecad6901 Two people have asked about section 1.3 problem 18e, so here are some general points to remember that might be helpful. * First point: The literal order in which quantifiers appear in an English sentence might not correspond exactly to the order they should appear in the corresponding logic translation. For example, consider the statement, "There is a person of every age." (For stating a person's age, we'll use the predicate A(x,y) where x is a person and y is an age.) Note that first we see the phrase "there is a person," and then we see "every age." You might think that this means that first we should have an EXISTS quantifier, followed by a FORALL quantifier: EXISTS x: FORALL y: A(x,y) However that is WRONG because it would mean: "There is a particular person such that for all ages, that person is that age." That would be nonsensical - a person can't have more than one age at a time! The problem is the order of the quantifiers. Given our commonsense knowledge, we know that what that particular sentence is actually trying to say is: "For every age, there is a person of that age." FORALL y: EXISTS x: A(x,y) Note that the correct order of the quantifiers (and the variables that go with them) is here the OPPOSITE of the order that the phrases "there is" and "every" appear in, in the orginal English sentence. This is all just a side effect of the inherent ambiguity of the English language, and illustrates why a formal mathematical language such as predicate logic is often preferable to natural human languages in mathematics: the mathematical language allows you to express complex concepts with complete precision and lack of ambiguity. * Second point: Remember that you can define predicates of any number of arguments, if it is convenient. For example, suppose I have already defined predicates A(x,y) = "person x is age y" and B(x,z) = "person x has profession z", and suppose I now want to say "Person x has age y and profession z." I could just write, A(x,y) /\ B(x,z). Or, for increased conciseness, I could define a new predicate C(x,y,z) = A(x,y) /\ B(x,z) = "person x has age y and profession z" and then I could just write C(x,y,z) instead. Inventing such substitutions is always something that you can do if you like, but of course it's not required. It just depends on which way you feel does the better job of communicating what you want to say. Feel free to email any other questions, or come to someone's office hours. Good luck, -Mike From mpf@cise.ufl.edu Tue Sep 7 10:19:39 1999 Date: Tue, 7 Sep 1999 10:19:39 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT 3100/7094X students) Subject: Web page for COT3100 X-UIDL: c7673411b92a7b66eb7bb9c40e29537f COT3100/7094X students, A reminder, and for those who were not in the first lecture: The web page for our section of COT3100 is http://www.cise.ufl.edu/~mpf/cot3100.html. There, you can view and print previous handouts, and check the most up-to-date versions of the course calendar and the staff office hours. Enjoy! -Mike Frank From mpf@cise.ufl.edu Fri Sep 3 15:03:28 1999 Date: Fri, 3 Sep 1999 15:03:28 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT 3100/7094X students) Subject: Correction X-UIDL: 4339709a84409b2680792edc30043f06 Oops.. I meant homework #2, not #1, of course. -Mike From mpf@cise.ufl.edu Fri Sep 3 15:02:49 1999 Date: Fri, 3 Sep 1999 15:02:49 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT 3100/7094X students) Subject: HW2 now due Friday 9/10 X-UIDL: 921288cd2c29a1a468d42608a83b0c42 Hello COT 3100 students; An important announcement: Since we didn't have time to finish covering Set Theory in lecture today, the due date for homework #1 is being pushed back to Friday (Sep. 10) rather than Wednesday (Sep. 8). However, I would recommend that you try to go ahead and do the exercises from sections 1.3 and 1.4 (Predicate logic and the first half of set theory) over the long weekend, while that material is still fresh in your mind, rather than waiting until after next Wednesday's lecture to do everything. I hope you all have a pleasant labor day weekend, and enjoy the football game (if you're going)! -Mike From mpf@cise.ufl.edu Wed Aug 25 21:59:33 1999 Date: Wed, 25 Aug 1999 21:59:33 -0400 (EDT) X-Authentication-Warning: rain.cise.ufl.edu: mpf set sender to mpf@rain.cise.ufl.edu using -f From: "Michael P. Frank" Subject: New COT3100 office hours. X-UIDL: a6b487d0b0c7dc19a9f285a586e8d961 Hello COT3100 6th period students, We have rearranged our office hours yet again, and established a new policy that you can visit the office hours of any of the 10 staff members from any of the 4 sections of COT3100. Almost every hour from 3rd to 11th period is covered every day by at least one staff member. To see the new schedule, visit http://www.cise.ufl.edu/~mpf/ohs.html. However, if you have a question about your grade on particular homework or test problems, please stick with the staff for our 6th period section, who will be grading your tests and assignments. Their hours are as follows: Name Email Hours Room ---- ----- ----- ---- Michael Frank mpf@cise.ufl.edu Wed. 5th CSE E442 (Instr.) Fri. 5th & 8th William Valella valella@ufl.edu Mon. & Tue. CSE E429 (TA) 7th & 8th Prajakta Ugrankar ugrankar@ufl.edu Wed. 7th & 8th CSE E429 (TA) Thu. 6th & 7th CSE E429 Enjoy! -Mike From mpf@cise.ufl.edu Wed Aug 25 15:51:14 1999 Date: Wed, 25 Aug 1999 15:51:14 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: knunery@hotmail.com (Kyle T. Nunery) Subject: Lost notebook, and new COT3100 office hours. X-UIDL: d541fc50cf23a1e7d088ba132e5d07f2 Hi COT 3100 students, This is a test of the class mailing list for our section. Please let me know if you don't get this message! :) Also, a student apparently left their spiral-bound, cover-less notebook in the classroom today (on the table by the door). If it's yours, come by my office (CSE E442) and get it. I said I would be holding office hours MWF 5th period, but TA William Valella (say va-LAY-ya, valella@ufl.edu) prefers to cover that time slot. He'll be in CSE room E429. (He has a class 4th period so he may be a few minutes late sometimes; just hang out and wait.) I will take MWF 8th period office hours, since I think that was the 2nd most popular time slot. Again please contact us if you need to arrange another time. Thanks for the feedback sheets... Keep them coming! -Mike Frank From mpf@cise.ufl.edu Tue Sep 21 14:25:37 1999 Date: Tue, 21 Sep 1999 14:25:37 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT 3100 students) Cc: pau@cise.ufl.edu, wvalella@cise.ufl.edu, xgu@cise.ufl.edu, eisland@ecel.ufl.edu Subject: COT3100 course news added to web page X-UIDL: 9a178ffacc4a2705c1b0ce8158799b7b The class web page http://www.cise.ufl.edu/~mpf/cot3100.html now begins with an "Announcements" section, which contains an archive of all these email announcements, and a summary of recent course news. -Mike From mpf@cise.ufl.edu Wed Sep 22 14:12:37 1999 Date: Wed, 22 Sep 1999 14:12:37 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT 3100 students) Cc: pau@cise.ufl.edu, wvalella@cise.ufl.edu, xgu@cise.ufl.edu, eisland@ecel.ufl.edu Subject: Extra office hours, rest of this week. X-UIDL: e07980c7fb458087f01c481c7aeac527 To assist students with learning material from section 1.8 which we did not have time to cover thoroughly in class today, and with reviewing material in preparation for the exam, I will be holding the following extra office hours, for this week only: Today (Wed. 9/22), 2:00pm - 5:00pm Tomorrow (Thu. 9/23), 11:00am - 1:00pm, 2:00pm - 6:00pm -Mike Frank From mpf@cise.ufl.edu Wed Sep 22 18:04:36 1999 Date: Wed, 22 Sep 1999 18:04:36 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT3100 students) Cc: pau@cise.ufl.edu, wvalella@cise.ufl.edu, xgu@cise.ufl.edu, eisland@ecel.ufl.edu Subject: Notes on Orders of Growth X-UIDL: abad0dc2e372f6082dc23378b9ef6e81 To review and supplement the book and lecture material on orders of growth, I have started writing some additional notes on the topic; these may be found at http://www.cise.ufl.edu/~mpf/order.html. Later I will also send out some homework hints for that section. -Mike From mpf@cise.ufl.edu Mon Sep 27 15:08:48 1999 Date: Mon, 27 Sep 1999 15:08:48 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT3100 students) Cc: pau@cise.ufl.edu, wvalella@cise.ufl.edu, sreddy@cise.ufl.edu, xgu@cise.ufl.edu, eisland@ecel.ufl.edu Subject: Crib sheet policy. X-UIDL: 74b4fd6c4fa9c51616272464bb2f3b8c Students, I have been neglecting to mention this lately, but I did say it in class one day, so I had better repeat it now, for benefit of those who might not remember: You are allowed to bring to the exam one 8 1/2" x 11" sheet of paper with whatever notes you want. (Front and back.) However, if you do this, I would strongly encourage you to make sure that you do also actually know everything that you write down, as it will save you much valuable time in the exam if you do not need to constantly look things up on the crib sheet. Good luck, -Mike From mpf@cise.ufl.edu Mon Sep 27 12:34:45 1999 Date: Mon, 27 Sep 1999 12:34:45 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT3100 students) Subject: Some general homework hints. X-UIDL: aa1ab2246d50e4bddd8b172e03c5602b Sorry I didn't get a chance to send out hints for section 1.8 earlier. This is too late to help anyone for HW3, but here are some general hints that may be useful in the future. As a general hint: The key to solving any new mathematical problem (discrete or otherwise) that you don't know how to approach is to take the symbols in the problem that you don't know what to do with, and eliminate them, by reducing them to their precise mathematical definitions, which will typically involve simpler concepts that are easier to work with. For example, in order to understand Omega(1) in problem 30, write down the literal definition of Omega(g) (either Definition 2 on p. 88, or the definition I gave in class). Then, substitute 1 for g(x) in that definition, and read what the resulting statement says, try to simplify it, and re-express what it says in your own words. Similarly, in problem 34, where you work with logarithms to different bases, think about how to break down log-base-a of x into a definition using simpler concepts. (You should know some identities for working with logarithms to different bases, from your algebra background.) I can't emphasize enough how important it is for you to learn this general skill... You can go a long way towards conquering a problem dealing with any new mathematical concept, if you simply take the time to refer back to the original definition of the concept, and then combine that definition with the statement of the problem you're given, by making substitutions in the appropriate places. Knowing to do this is one of the most fundamental mathematical skills. Also, it's an easy thing to do, once you are in the habit of doing it. But apparently, it doesn't occur to a lot of people to just go back to the definitions, and work from them. Good luck, -Mike From mpf@cise.ufl.edu Mon Sep 27 10:52:28 1999 Date: Mon, 27 Sep 1999 10:52:28 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT3100 students) Cc: pau@cise.ufl.edu, wvalella@cise.ufl.edu, sreddy@cise.ufl.edu, xgu@cise.ufl.edu, eisland@ecel.ufl.edu Subject: Review session. X-UIDL: 2c6432af372ced80791da270246f3307 The exam review session will be Monday (9/27), 10th-11th periods (5:10pm - 7:05pm) in CSE room E119. Please come if you can. I will be available after the main review session, to answer additional questions. I have finished formatting the sample exam problem solutions. The problems and solutions can be found, respectively, at: http://www.cise.ufl.edu/~mpf/exam1-samp.html http://www.cise.ufl.edu/~mpf/exam1-sampsol.html Reminder: HW3 is due today! -Mike From mpf@cise.ufl.edu Fri Sep 24 14:58:29 1999 Date: Fri, 24 Sep 1999 14:58:29 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT3100 students) Cc: pau@cise.ufl.edu, wvalella@cise.ufl.edu, sreddy@cise.ufl.edu, xgu@cise.ufl.edu, eisland@ecel.ufl.edu Subject: Solutions to exam 1 sample problems. X-UIDL: 6ce66e1f8045e194f3815193b2025e6b At the following web page are solutions to the exam sample problems. As of now, I have not yet finished formatting all of these in HTML, but hopefully you can still read the last three solutions. I will probably have them all formatted by Monday. http://www.cise.ufl.edu/~mpf/exam1-sampsol.html I would recommend you try solving the problems yourself first without looking at the solutions, and then look at them afterwards to check your work. Enjoy, -Mike From mpf@cise.ufl.edu Thu Sep 23 11:59:29 1999 Date: Thu, 23 Sep 1999 11:59:29 -0400 (EDT) X-Authentication-Warning: frond.cise.ufl.edu: mpf set sender to mpf@frond.cise.ufl.edu using -f From: "Michael P. Frank" To: (COT3100 students) Cc: pau@cise.ufl.edu, wvalella@cise.ufl.edu, sreddy@cise.ufl.edu, xgu@cise.ufl.edu, eisland@ecel.ufl.edu Subject: HW3 & Exam #1 pushed back. X-UIDL: 4295e7704b23129d8c5a5ab4ee5d4dd3 Students, I apologize for all of the recent reschedulings, but I am hereby pushing back the due date for HW3, for the third time, to next Monday (Sep. 27), so that we have time to cover the material on orders of growth more thoroughly. The exam is also being rescheduled, to next Wednesday (Sep. 29), so that you will have time to study for it after turning in the homework. I am sorry if the new time inconveniences you, but I feel it is necessary. I am still planning to have an exam review session, but it might be held Monday now, instead of tomorrow. I am also planning to give out exam practice questions as soon as I can put them together. Apologies again for all of the disruptive changes... I believe our original schedule for covering chapter 1 was over-ambitious. It's important that we try to cover these fundamentals thoroughly, before moving on to more advanced topics that build on them. Regards, -Mike