| SCHEDULE | |
| ASSIGNMENTS | |
| REVIEW SHEETS: First Exam, Second Exam, Third Exam, Fourth Exam, Final Exam, |
This course serves two main populations, students from mathematics and students from computer science. The course also has two distinct goals. One is to teach the basics of set theory, logic, combinatorics and graph theory. The other is to convey the rigorous use of terminology and proof which is essential to mathematics: that is, clarity and precision in definitions and statements of fact, and logical methods for establishing that a statement is true. The fundamental mathematics taught in this course is useful in computer science, electrical engineering. The analytical skills developed are critical to understanding computer languages, to the development of good programming skills, and, more generally, to appreciating scientific method. The experience with abstract concepts, proof writing and the use of formal definitions should help students in all disciplines to articulate their ideas more clearly.
| SECTIONS | TOPICS | TIME |
| §1.1-4 | Logic and logical arguments. | 4 classes |
| §2.1-3 | Predicates and quantifiers. | 4 classes |
| §3.1-7 | Proof: direct proof, division into
cases,proof by contraposition, proof by contradiction and disproof by counterexample. |
6 classes |
| Some number theory:
Statements of the division theorem, unique factorization.
Proof of the irrationality of sqrt(2), proof of the infinitude of primes. |
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| §5.1-3 | Sets: subsets, union, intersection. | 4 classes |
| Venn diagrams. Algebra of set operations. | ||
| Cartesian product, power set. | ||
| §10.1-3,5 | Relations. Reflexive, symmetric and transitive relations. | 6 classes |
| Equivalence relations. Partially ordered sets | ||
| §7.1,3,4,5 | Functions: one-to-one and onto functions. | 4 classes |
| Invertible functions. The pigeonhole principle. | ||
| Composition of functions. | ||
| §4.1-4 | Sequences, mathematical induction | 4 classes |
| §8.1-3 | Recursively defined sequences: | 4 classes |
| Finding explicit formulas, | ||
| establishing the formulas by induction. | ||
| §6.1-7 | Combinatorics: Counting and the multiplication rule. | 6 classes |
| Permutations and combinations. |
We will have weekly exams (5 at 100 points each), weekly homeworks (6 at 50 points each). I will also assign 200 points based on class participation, quizes, etc. For the weekly assignments (see above), there will be a small number of problems (10 or so) which you should write up carefully. They will be graded and returned to you promptly (to the best of my ability).
Most students find this a tough class. You will greatly magnify your chances of succeeding if you work regularly and attentively! Start the assignments early and do most of the exercises in each section we cover. If you get stuck, please ask me in class. Your questions often lead to a valuable class discussion. You are also quite welcome during my office hours, but come prepared.