Course Description and Goals
From the graduate
bulletin: Fundamental Concepts of Mathematics. Brief history of
mathematics; sets, functions and logic; constructions of number systems;
mathematical induction. The main focus of the course will be on the
construction and writing of mathematical proofs.More specifically, students completing MAT 511 should be able to
- be able to construct logically correct sentences using predicates, connectives and quantifiers;
- be able to construct useful denials of propositions involving connectives and quantifiers;
- understand and use literately terminology associated with conditional and biconditional sentences (sufficient and necessary conditions, antecedent and consequent, etc.) and understand the difference between implication in mathematics and causation in everyday life;
- formulate and prove de Morgan’s Laws;
- understand the structure of mathematical definitions and theorems;
- understand and use fluently various proof techniques: direct proof, proof by contraposition, proof by contradiction, proof by cases;
- be able to identify, analyze and prevent typical logical mistakes, like affirming the consequent, denying the antecedent, etc.;
- be able to apply the principle of mathematical induction in problem solving;
- know the basic language of the set theory (membership, union, intersection, the empty set, complement) and be able to calculate the complement of a set defined in those terms;
- be able to relate operations of logic and set theory (e.g, negation and complement, conjunction and intersection, etc.);
- be able to prove simple set-theoretical identities and illustrate them by Venn diagrams;
- be familiar with the power set and indexed family of sets;
- understand the definition of direct product of sets;
- be familiar with the terminology associated to binary relations (e.g., reflexive, irreflexive, symmetric, antisymmetric, and transitive relations);
- be familiar with strict and non-strict partial orders;
- be able to test whether a relation is an equivalence relation;
- operate with equivalence classes and the quotient set, and understand the connection between partitions of a set and equivalence relations on the set;
- know the definition of a function and use literately the terminology related to functions and mappings (e.g. domain, codomain, range, image, pre-image);
- be familiar with special types of maps like identity maps, constant maps, restrictions, projections, inclusions, factor-maps, submaps, the characteristic function of a set;
- know the definition of a metric on a set and be able to recognize a metric;
- know the definition of injective, surjective and bijective map, and determine if a map is or is not injective, surjective, bijective;
- know the definition of an invertible map and its reformulation, and be able to check if a map is or is not invertible;
- be familiar with the notion of cardinality and related notions (finite and infinite sets, denumerable sets);
- be familiar with the axiomatic construction of natural numbers and the Peano axioms, and be able to prove equivalence of principle of mathematical induction and well ordering principle;
- use the Pigeonhole principle in proofs and problem solving;
- be familiar with basic results about countability (e.g., Q are denumerable, R is uncountable);
- be able to formulate the Cantor-Schröder-Bernstein theorem and prove that cardinal numbers are linearly ordered.
Textbook
 |
Textbook: A
transition to advanced mathematics Smith, Eggen and St Andre. Cengage Learning |
Lectures
New material is presented each week in the lectures. You should read the
corresponding topics beforehand.Grading System
The relative weight of exams and problem sets in determining final grades is as follows. Class participation, in Lectures and Recitations, will count in your favor if your grade is "between" two letters.|
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% |
When and where | |
|---|---|---|---|
|
|
20% | Thursday Oct 16 in class. | |
|
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40% | Tuesday 12-16, P-131 at 5:30pm | |
|
|
20% | Weekly |
|
| Class
Participation |
20% | Daily, in class! |
If you miss a midterm for a documented
reason, the score will be replace by the grade on the balance of the
course. No make-ups will be given.
Written Homework
Homework will be assigned weekly and posted on the Homework
webpage. The homework must be turned to your instructor and will be due
at the second meeting of your class the following week. Only the
problems indicated in bold are to be turned in. You should still do all
problems assigned.No late homework will be accepted.
All problem sets handed stapled. Moreover, problems must be legible and must use complete sentences, correct grammar, correct spelling, etc. Problem sets which prove too difficult for the grader to read may be marked incorrect or may be returned to the student for rewriting (as the instructor sees fit). A complete solution will include the following:
- The statement of the problem
- An organized presentation of ideas leading to a solution
- An answer that is circled or boxed
- If a problem has multiple parts it should be solved as though each part were a separate problem, following the order in which parts are listed.
- If there is no work shown, there is no credit. In other words, an answer with no justification is not admissible (even if it is the correct answer!)
We encourage you to form teams of three or four students and to work together. We will try to do as many group exercises as possible, in class and in recitation, to get you used to this type of work. Several people thinking together about a problem can often see around a difficulty where one person might get stuck. (This is one reason why the ability to work well in a team is rated very highly by prospective employers. ) Please note that even if students are encouraged to discuss and work on the problems with your team, the final write-up must be individual.
If you do not understand how to do something, get help from your instructor, your, your classmates, in the Math Learning Center or at the Stony Brook Tutoring Center. You are encouraged to study with and discuss problems with others from the class, but write up your own homework by yourself, and make sure you understand how to do the problems.
Never be shy to ask us how to do a homework problem, even if you handed in solution that you do not understand. We will be glad to help you!.
Graded problem sets and exams will be handed back in recitation. If you cannot attend the recitation in which a problem set or exam is handed back, it is your responsibility to attend your recitation instructor's office hours and get your graded work. Failure to retreive graded work is not grounds for a make-up, a regrade or change of a fail to an incomplete.
You are responsible for collecting any graded work by the end of the semester. After the end of the semester, the recitation instructor is no longer responsible for returning your graded work. If you have a question about the grade you received on a problem set or exam, you must contact the recitation instructor (not the grader or the lecturer).