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CS210: Discrete Mathematics

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 1 / 12

Goals

Main goals of this course are learning problem-solving skills fundamental structures in computer science to think clearly and precisely One of the few problem-solving courses you’ll take at LUMS. Sets the foundation for various courses such as ‘Data Structures’, ‘Automata and Complexity’, ‘Algorithms’, ‘Compilers’, ‘Vision’, ‘Networks’, ‘Operating Systems’.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 2 / 12

Course Outline

“Invitation to Discrete Maths” 2nd ed., Matousek & Nevestril Logic & Proofs: Different kinds of Proofs, Induction. Analysis Tools: Summations, Asymptotics, Recurrences. Combinatorics: Permutations, factorials and binomial coefficients, Inclusion-exclusion, Derangements, Double-counting techniques, Parity arguments. Graphs: Isomorphism, representation, degree sequences, Euler tours, 2-connectivity, chromatic number and matchings. Trees: Properties, spanning trees, computing minimum spanning trees. Number Theory: Linear combinations, fundamental theorem, congruences, mult. inverse existence, Euler’s theorem, RSA. Probability: Birthday paradox, Random graphs, Random variables, Expectation, Linearity of expectation, Markov’s inequality, Variance, Chebychev’s inequality.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 3 / 12

Grading

Quizzes: 20%

Five quizzes in the course, each worth 4% of the total. Time for each quiz is 30 minutes. Open book. Open notes.

Homeworks: 20%

There will be a homework every week. Each homework has 4-6 questions. Homeworks can be done in groups of 3 students.

Midterm: 20%, Final: 40%

There will be 4-5 questions in each exam. Allowed time: 4 hours. Open book. Open notes.

There’s an absolute grade: below 20% aggregate is ‘F’.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 4 / 12

Problem-solving Tutorials

Every week, there will be a 2-3 hour problem-solving tutorial. Myself, and the TAs will prepare several problems.

In the tutorial, we’ll discuss these problems. Discuss your solution and present our solutions.

Bring your own practice questions, everyone can try them. You can ask anything you didnt understand in the lectures. Homeworks and quizzes will be solved, and discussed. Participation is voluntary. Time for these tutorials: Friday 10:30am-12:30pm

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 5 / 12

Administrative Details

Instructor: Nabil Mustafa

Room 411. Office Hours: 2:30-3:30pm, Tuesday/Thursdays. Email: nabil@lums.edu.pk

TAs:

Mirza Basim Baig, 11020448@lums.edu.pk Mariyam Khalid, 11020252@lums.edu.pk

DM is a core courses in the CS curriculum

If you do badly in DM, difficulty in several later CS courses.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 6 / 12

Cheating

Homeworks

Homeworks can be done in groups of three students. Can discuss problems and solutions with your groupmate, but have to write individual solutions. Can discuss problems and solutions with the TAs or myself. Cannot look up solutions on the internet. Cannot discuss problems or solutions with other classmates.

Any cheating case we suspect will be forwarded to the disciplinary committee. No second chances. Homeworks to be handed in at the beginning of the Tuesday

• class. Late homeworks are not accepted.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 7 / 12

Some Advice

“I don’t have any magical ability. I look at the problem, and it looks like one I’ve already done. When nothing’s working out, then I think of a small trick that makes it a little better. I play with the problem, and after a while, I figure out what’s going

• n.”

–Terence Tao, Fields medal for combinatorics

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 8 / 12

Some Advice

Good advice is always certain to be ignored, but that’s no reason not to give it. –Agatha Christie

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 9 / 12

Some Advice

Good advice is always certain to be ignored, but that’s no reason not to give it. –Agatha Christie Get comfortable reading and understanding the textbook – even if its rather painful initially.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 9 / 12

Some Advice

Good advice is always certain to be ignored, but that’s no reason not to give it. –Agatha Christie Get comfortable reading and understanding the textbook – even if its rather painful initially. You could be stuck on a question for hours. That is normal, and you should not panic at that.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 9 / 12

Some Advice

Good advice is always certain to be ignored, but that’s no reason not to give it. –Agatha Christie Get comfortable reading and understanding the textbook – even if its rather painful initially. You could be stuck on a question for hours. That is normal, and you should not panic at that. Often you will get insight into the questions at odd times. Try to keep thinking about the problem continuously and patiently.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 9 / 12

Some Advice

Good advice is always certain to be ignored, but that’s no reason not to give it. –Agatha Christie Get comfortable reading and understanding the textbook – even if its rather painful initially. You could be stuck on a question for hours. That is normal, and you should not panic at that. Often you will get insight into the questions at odd times. Try to keep thinking about the problem continuously and patiently. Grading is more absolute than relative. So worry if you are getting low scores, even if you’re above the mean.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 9 / 12

Some Advice

Good advice is always certain to be ignored, but that’s no reason not to give it. –Agatha Christie Get comfortable reading and understanding the textbook – even if its rather painful initially. You could be stuck on a question for hours. That is normal, and you should not panic at that. Often you will get insight into the questions at odd times. Try to keep thinking about the problem continuously and patiently. Grading is more absolute than relative. So worry if you are getting low scores, even if you’re above the mean. Both TAs are very hard-working intelligent students (they were the only two students who got an ‘A+’ grade last year). Discussing solutions/ideas with them will really help!

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 9 / 12

Some Advice

Good advice is always certain to be ignored, but that’s no reason not to give it. –Agatha Christie Get comfortable reading and understanding the textbook – even if its rather painful initially. You could be stuck on a question for hours. That is normal, and you should not panic at that. Often you will get insight into the questions at odd times. Try to keep thinking about the problem continuously and patiently. Grading is more absolute than relative. So worry if you are getting low scores, even if you’re above the mean. Both TAs are very hard-working intelligent students (they were the only two students who got an ‘A+’ grade last year). Discussing solutions/ideas with them will really help! Ask specific questions, as they are easier to answer. General questions like ‘Can you explain Lecture 11 again?’ or ‘What is the meaning of life’ will get you little help.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 9 / 12

Some More Advice

We all admire the wisdom of people who come to us for

• advice. –Jack Herbert

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 10 / 12

Some More Advice

We all admire the wisdom of people who come to us for

• advice. –Jack Herbert

Understanding the solution to a problem, and discovering the solution are two entirely different things. Very often you will struggle with a question for hours, and then be amazed at the simplicity of the solution. It happens to everyone, so don’t be discouraged by that. It’s what makes math beautiful.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 10 / 12

Some More Advice

We all admire the wisdom of people who come to us for

• advice. –Jack Herbert

Understanding the solution to a problem, and discovering the solution are two entirely different things. Very often you will struggle with a question for hours, and then be amazed at the simplicity of the solution. It happens to everyone, so don’t be discouraged by that. It’s what makes math beautiful. Often students say ‘I understood everything in class, but am unable to solve problems’. The secret is practice, practice and more practice. No one else can really teach you mathematics.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 10 / 12

Some More Advice

We all admire the wisdom of people who come to us for

• advice. –Jack Herbert

Understanding the solution to a problem, and discovering the solution are two entirely different things. Very often you will struggle with a question for hours, and then be amazed at the simplicity of the solution. It happens to everyone, so don’t be discouraged by that. It’s what makes math beautiful. Often students say ‘I understood everything in class, but am unable to solve problems’. The secret is practice, practice and more practice. No one else can really teach you mathematics. One of the great advantages of objective fields like mathematics is that looks don’t matter. There’s no need to ‘appear smart’. In class, concentrate and ask whenever you get confused. It is much better to look dumb and do well in exams than the other way around.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 10 / 12

Some More Advice

We all admire the wisdom of people who come to us for

• advice. –Jack Herbert

Understanding the solution to a problem, and discovering the solution are two entirely different things. Very often you will struggle with a question for hours, and then be amazed at the simplicity of the solution. It happens to everyone, so don’t be discouraged by that. It’s what makes math beautiful. Often students say ‘I understood everything in class, but am unable to solve problems’. The secret is practice, practice and more practice. No one else can really teach you mathematics. One of the great advantages of objective fields like mathematics is that looks don’t matter. There’s no need to ‘appear smart’. In class, concentrate and ask whenever you get confused. It is much better to look dumb and do well in exams than the other way around. Taking long solitary walks at night can really help in thinking.

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 10 / 12

I Hope I Never See Things Like These

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 11 / 12

I Hope I Never See Things Like These

Nabil H. Mustafa http://russell.lums.edu.pk/~cs210f09 CS210: Discrete Mathematics 12 / 12