The right way
to teach and learn math -- my new book: Rediscovering
Mathematics

**Text: ** Discrete
Mathematics
and its Applications, Rosen, McGraw Hill, 7th edition.

This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis will be placed on providing a context for the application of the mathematics within computer science. The analysis of algorithms requires the ability to count the number of operations in an algorithm. Recursive algorithms in particular depend on the solution to a recurrence equation, and a proof of correctness by mathematical induction. The design of a digital circuit requires the knowledge of Boolean algebra. Software engineering uses sets, graphs, trees and other data structures. Number theory is at the heart of secure messaging systems and cryptography. Logic is used in AI research in theorem proving and in database query systems. Proofs by induction and the more general notions of mathematical proof are ubiquitous in theory of computation, compiler design and formal grammars. Probabilistic notions crop up in architectural trade-offs in hardware design.Description:

**Exams:** There will be 6
quizzes (40% total) and one final examination (35%).** **Each
quiz will be worth 20 points and the lowest quiz grade will be
dropped**. **Quizzes will be taken in discussion
sections and administered and graded by the TAs.**
**

Teaching Assistants:The teaching assistants will grade your homework assignments and quizzes, and answer questions at labs under my guidance and with my instructions. If you have a question or complaint about a HW or quiz grade, and you are unable to resolve it with the TA, please see me for help.

Discussion Sections:The TAs will alternate between giving quizzes, and reviewing. You are expected to attend the discussion section for which you are signed up. The TAs will take attendance and it will be counted in your HW grade.The teaching assistants this semester are:

**Assignments:** Homework
assignments will be worth 25% of your grade (5% each
assignment). The HWs are meant to force you to study
and practice. If you work hard and meet with the TAs and
myself, you can always be sure to work through every problem and
get full credit. You may work with a partner on the homework and
submit one assignment for both of you. If you choose to do
this, then stick together all semester. It is easier to work
with a partner but you will get less practice.

**Grades: **I count your HWs
and lab attendance for 25%, quizzes for 40%, and the final as
35%. You can guarantee an A, B, C, D grade with at least a
90, 80, 70, 60 percent respectively. I may lower these
numbers (in your favor) at the end of the semester if I feel a
curve is justified.

**Academic Honesty: ** I value
hard work and honest effort. If you are working with a
partner, make sure the TAs know you are submitting work as a
group. If you use any resource on the Internet to help you
with your HW, or you get a hint from a fellow student or TA, you
must rewrite the idea in your own words and reference your
source. Copying work or rewriting it in your own words
without any reference has potential serious consequences,
including failing the course. Moreover, this kind of
dishonesty will not prepare you properly for the quizzes and
final. If a TA reports that your HW violates academic
integrity guidelines, I will take a look and if I agree, then you
will get a zero on that assignment. If it happens twice,
then I will follow the school's more serious policies.
Read my academic
integrity guidelines before you hand in any written work.

**Goals:** To understand the
mathematics that underlies computer science, and to appreciate
where it is used.

** Special
Dates:** April 16 and 21 are Jewish holidays
and I will not lecture on those dates. There will be a
substitute lecturer, or possibly a quiz.

- First week: Read This First to learn how to read mathematics.
- First month: Solve this puzzle online here.
(Works best with Internet Explorer).

Written Assignments

Note:
These are long assignments so you should spread your efforts
over a period of 2-3 weeks for each one; you will learn
more and get better grades if you do. The TAs will review
any questions you have, and I will give hints and guide
you. Never give up.

Assignment 1 |
Assignment 2 |
Assignment 3 |
Assignment 4 |
Assignment 5 -
Due last day of class |

Week | Topics | Reading |

1 | What kinds of problems are solved
in discrete math? What are proofs? Examples of proofs by
contradiction, constructive proofs, and proofs by induction:
Triangle numbers, irrational numbers, and prime numbers. |
1.7-1.8, 5.1 |

2 | Primes, Greatest Common Divisors
and the Euclidean Algorithm. Binary numbers and
conversions from binary to decimal and vice versa. The
Egyptian fast-exponentiation algorithm. The two-jug
puzzle as demonstrated by Bruce Willis in Die Hard III.
Congruences and Fermat’s Little theorem. Applications to
Cryptography. |
4.1-4.6 |

3 | Mathematical induction – a flexible and useful tool. Many examples and the idea of strong induction. | 5.1-5.2 |

4 | Basic arithmetic and geometric sums, closed forms. Growth rate of functions, Big-O notation. Applications to algorithms. | 2.3-2.4, 3.1-3.3 |

5 | Recursion and solving recurrence equations by repeated substitution: Compound Interest, Binary Search, Insertion Sort, Merge Sort. Towers of Hanoi – graphs as a visual tool. |
5.3-5.4 |

6 | Recursion and solving recurrence equations by repeated substitution: Chinese rings puzzle – Grey codes, change of variable technique. |
702-703 |

7 | Solving recurrence equations: Master Theorem with applications to algorithms. Guessing and proving correct by induction - The Josephus Problem, and kth largest. |
8.1, 8.3 |

8 |
Solving recurrence equations: Linear homogeneous equations, linear non-homogeneous equations. |
8.2 |

9 |
Combinations and permutations. Pascal’s triangle and binomial coefficients. |
6.1, 6.3, 6.4 |

10-11 | Counting problems using combinations, distributions and permutations. |
6.5-6.6 |

12 |
Discrete
probability, the birthday paradox, conditional
probability, expected value, and many examples. |
7.1-7.4 |

13-14 | Set Theory - Inclusion/Exclusion Theorem, Associative, Distributive, and De Morgan's Laws. Bit Operations in Java. Functions, 1-1 correspondence, and Countability - Diagonalization. Applications to Computability and Undecidability. |
1.1-1.3, 2.1-2.5 |

15 |
Propositional Logic and Boolean Algebra, Logic Gates, Circuits and Applications | 12.1-12.3 |