Course Meeting Times

Lectures: 3 sessions / week, 1.5 hours / session



This subject offers an introduction to Discrete Mathematics oriented toward Computer Science and Engineering. The subject coverage divides roughly into thirds:

  1. Fundamental concepts of mathematics: definitions, proofs, sets, functions, relations.
  2. Discrete structures: graphs, state machines, modular arithmetic, counting.
  3. Discrete probability theory.

The prerequisite is 18.01 (first term calculus), in particular, some familiarity with sequences and series, limits, and differentiation and integration of functions of one variable.

The goals of the course are summarized in a statement of Course Objectives and Educational Outcomes.

Considerations for Taking the Subject This Term

There are two main considerations for students in deciding to take 6.042J/18.062J this term—or at all.

  1. Team Problem Solving
  2. This term, as in many previous terms, the subject is being taught in Lecture/Team Problem Solving style. More than half the class meeting time will generally be devoted to problem solving in teams of 7 or 8 sitting around a table with a nearby whiteboard where a team can write their solutions. Each TA covers 3 tables, acting as coach and providing feedback on students' solutions. The Lecturer acts likewise, circulating among all the tables. The coach will resist answering questions about the material, always trying first to find a team member who can explain the answer to the rest of the team. Of course the coach will provide hints and explanations when the whole team is stuck.

    Problem solving sessions will generally be preceded by half hour presentations by the lecturer, usually reviewing just the topics needed to understand the problems. These overviews are not intended as first-time introductions to the material nor as complete coverage of the assigned reading.

    The Good News is that the immediate, active engagement in problem solving sessions is an effective and enjoyable way for most students to master the material. Team sessions also provide a supervised setting to acquire and practice technical communication skills. Student grades for problem solving sessions depend on degree of active, prepared participation, rather than problem solving success. Sessions are not aimed to test how well a student can solve the problems in class; the goal is to have them understand how to solve them by the end of the session. Participation in team sessions counts for 20% of the final grade.

    In-class team problem solving works to solidify students' understanding of material they have already seen. The Bad News is that this requires students to arrive prepared at the sessions: they need to have read (though not carefully studied) the assigned reading and done the Online Tutor problems before class. There is no way to make up for not working with the team, so it is necessary to keep up and be there—no focusing on something else for a month, aiming to catch up afterward. If you cannot commit to keeping up, you may prefer to take the subject some other term.

  3. This subject covers many mathematical topics that are essential in Computer Science but are not covered in the standard calculus curriculum. In addition, the subject teaches students about careful mathematics: precisely stating assertions about well-defined mathematical objects and verifying these assertions using mathematically sound proofs. While some students have had earlier exposure to some of these topics, in most cases they learn a lot more in 6.042J/18.062J.

The subject is required of all Computer Science (6-3) majors and is in a required category for Math majors taking the Computer Science option (18C). But students with a firm understanding of sound proofs, and who are familiar with many of the covered topics, should discuss substituting a more advanced Math subject for 6.042 with the Lecturer or their advisor.

Problem Sets

Problem Sets are normally due at the beginning of lecture on Fridays, but a few may be due at alternate times because of holidays. Doing the problem sets is, for most students, crucial for mastering the course material. Solutions to the problem sets will be posted immediately after the due date. Consequently, late problem sets will not be accepted.

Problem sets count for 25% of the final grade.

Students are encouraged to collaborate on problem sets as on teams in class. The last page of each problem set has a collaboration statement to be completed and attached as the first page of a problem set submission:

"I worked alone and only with course materials"


"I collaborated on this assignment with (students in class),

got help from (people other than collaborators and course staff),

and referred to (citations to sources other than the material from this term)".

No problem set will be given credit until it has a collaboration statement.

Online Tutor Problems

There are weekly Online Tutor problems due before class on specified dates. These consist of straightforward questions that provide useful feedback about the assigned material. Tutor problems should take about 20 minutes after the reading has been completed. (Some students prefer to try the tutor problems before doing the reading, which is fine.)

Like team problem-solving in class, online tutor problems are graded solely on participation: students receive full credit as long as they try the problem, even if their answer is wrong. Tutor problems count for 5% of the final grade.

Weekly Reading Comments

A comment on the week's reading is due on specified dates by 9AM before class.

A single comment citing some paragraph that specially catches your attention is all that is required. The comment should indicate why this paragraph stood out, for example, because you found it especially

  • difficult/confusing, or
  • surprising, or
  • mistaken (pointing out typos & suggesting corrections is appreciated), or
  • funny, or
  • boring, or
  • lacking Computer Science motivation, or
  • poorly written,
  • something you'd like reviewed in class, ....

Comments in response to other people's comments are generally also OK to satisfy this requirement. Multiple comments are welcomed.

Note that global comments such as "I understood everything in the reading, found it all interesting, and have no questions" are not considered responsive. Even if you understood everything, there must, in the 15 to 30 pages assigned each week, have been something that stood out for you as suggested above. Similarly, responses to other comments such as "I agree," "lol,"..., are not sufficient.

Reading comments count for 5% of the final grade.

Collaboration and Outside Sources

We encourage students to collaborate on homework as on in-class problems. Study groups can be a big help in mastering course material, besides being fun and a good way to make friends. However, students must write up solutions on their own, neither copying solutions nor providing solutions to be copied. All collaborators must be cited, and if a source beyond the course materials is used in a solution—for example, an "expert" consultant other than 6.042 staff, or another text—there must be a proper scholarly citation of the source.

We discourage, but do not forbid, use of materials from prior terms other than those available on OCW. We repeat, however, that use of material from any previous term requires a proper scholarly citation. As long as a student provides accurate citation and collaboration statements, a questionable submission will rarely be sanctioned—instead, we'll explain why we judge the submission unsatisfactory (and maybe deny credit for specific, clearly copied solutions). But omission of such a citation will be taken as a priori evidence of cheating, with unpleasant consequences for everyone.

Biweekly Mini-quizzes

A 25-30 minute mini-quiz will generally be given every other week, usually on Wednesdays. Mini-quizzes count for a total of 20% of the final grade.

Material to study for a mini-quiz is very well defined: a mini-quiz will cover only the material in problems from the previous two weeks. Mini-quiz questions are often simply some parts of these online, class, and problem set problems. Students can prepare for a mini-quiz simply by reviewing the posted problem solutions for the previous two weeks.

Final Exam

There will be a standard 3 hour final exam. This exam is worth 25% of the final course grade.


The lowest mini-quiz score and problem set score, and the lowest two team problem-solving scores will not count in grade calculation. This effectively gives everyone 1 mini-quiz, 1 problem set, and 2 team problem-solving sessions they can miss without excuse or penalty.

Grades for the course will be based on the following weighting:

Problem sets 25%
Final 25%
Class participation 20%
Mini-quizzes 20%
Weekly reading comments 5%
Online tutor problems 5%

LaTeX Macros

Course handouts are formatted using LaTeX, which is the preferred formatting system among Mathematics professionals. Note that we do not think it's worthwhile for students to use it for their class submissions.