Math 109 -- Mathematical Reasoning -- Spring 2019

Lecture site: APM B402A
Lecture times: Tuesday, Thursday. 3:30pm-4:50pm.
Discussion sessions E01 970537: Friday 5:00p-5:50p, AP&M B412 with Alexander Mathers
E02 970538: Friday 6:00p-6:50p, AP&M B412 with Nandagopal Ramachandran
Final Exam
time and place
June 10th, 2019, 3:00pm - 6:00pm. Place TBA
Instructor Martin Licht
Email: mlicht AT ucsd DOT edu
Office: AP&M 5880E
Hours: Wednesday. 2:00 - 4:00pm. May be subject to change during the quarter.
Teaching Assistant Alexander Mathers
Email: amathers AT ucsd DOT edu
Office Hours: Mondays 4-5pm, Thursday 11:30am-12:30pm, AP&M 5412
Teaching Assistant Nandagopal Ramachandran
Email: naramach AT ucsd DOT edu
Office Hours: Wednesdays and Thursdays 5-6pm, AP&M 6436
Sections 970537, 970538
Credit Hours: 4 units
Course content This course uses a variety of topics in mathematics to introduce the students to rigorous mathematical proof, emphasizing quantifiers, induction, negation, proof by contradiction, naive set theory, equivalence relations and epsilon-delta proofs. Required of all departmental majors.
Formal prerequesite Math 18 or Math 20F or Math 31AH, and Math 20C. Students who have not completed listed prerequisites may enroll with consent of instructor.
Homework information Homework will announced on Fridays. Homework must be submitted to the mailboxes on Friday's before noon. No homework will be accepted in class. All submissions must be in handed-in in handwritten form.
Homework Philosophy The homework is the most important part of the course. Along with more straightforward problems designed to solidify the basic definitions and concepts, the homework will contain some problems which you might find difficult and are meant to challenge you. Don't worry if you get stuck; the process of thinking hard about a problem and trying different ideas is extremely valuable in itself and important in developing mathematical maturity. Of course, working through the homework is also vital for exam preparation.
Collaboration Policy You are welcome to discuss lecture material and homework problems with other students at the stage when you are still formulating ideas. This may be especially useful if, for example, you are confused about definitions or what the problem is asking. The write-up you hand in should be your work alone in your own words, however, and should be written while you are by yourself. While it is also fine to seek hints from classmates that have figured out problems on which you are stuck, you will learn the most if you think about these problems hard on your own first and don't give up too quickly.
Academic Integrity Every student is expected to conduct themselves with academic integrity. Violations of academic integrity will be treated seriously. See for UCSD Policy on Integrity of Scholarship.
Resources recommended
in addition

Grading Information

The final grade will composed 20% homework and either (a) 20% of the first midterm, and 20% of the second midterm, and 40% of the final exam or (b) 20% of the best midterm and 60% of the final exam, depending on what gives the better result.

Your course grade will be determined by your cumulative average at the end of the quarter, based on the following scale:

A+ A A- B+ B B- C+ C C-
100 - 96.66 96.65 - 93.33 93.32 - 90.00 89.99 - 86.66 86.65 - 83.33 83.32 - 80.00 79.99 - 76.66 76.65 - 73.33 73.32 - 70

The above scale is guaranteed: for example, if your cumulative average is at least 73, then your final grade will be at least B. However, your instructor may adjust the above scale to be more generous.

Course Calendar

Lecture Content
# 1, 1T
Administrativa. Examples and Motivation.
Formal logic: Statements, Implications, Equivalences. Truth Values.
# 2, 1Th
Formal logic: Propositions and Logical operators: and, or, negation, either-or. Truth tables. Further logical connectives. Composition of logical operations.
# 3, 2T
Set theory: Naive Set Theory. Set-theoretical operations. Definition of Sets. Set comprehension. Power sets.
# 4, 2Th
Set theory: Union, intersection, difference. Relations to logical operations and connectives.
Quantifiers: Existence and Universal Quantifiers.
# 5, 3T
Set theory: Infinite Unions and Intersections. Cardinality. Proofs involving set theory.
# 6, 3Th
Proofs in combinatorics: Some examples for proofs in combinatorics. Direct proofs. Proofs by induction.
#7, 4T
Proofs in combinatorics: Proof by induction in combinatorics.
#8, 4Th
First Midterm
Set theory: Classes.
26.04.2019: Deadline to change grading option, change units, and drop classes without "W" grade on transcript.
#9, 5T
Elementary number theory: well-ordering principle and divisibility. division-remainder theorem, prime numbers, greatest common divisor.
#10, 5Th
Elementary number theory: Euclidean algorithm, Bezout's lemma. Infinitiy of the number of prime numbers. Fundamental theorem of arithmetics. Applications.
#11, 6T
Elementary number theory: Mersenne prime numbers. Mersenne prime numbers. Two examples for mathematical induction.
#12, 6Th
Second Midterm.
10.05.2019: Deadline to drop with "W" grade on transcript.
#13, 7T
Further Set Theory: Ordered Pairs and Cartesian Products. Functions. Graphs. Injectivity, Surjectivity, Bijectivity.
#14, 7Th
Further Set Theory: Tuples and Sequences. Relations.
#15, 8T
Further Set Theory: Equivalence Relations. Orders.
#16, 8Th
Further Set Theory: Total Orders. Lexicographical ordering on Cartesian Products. Operations on Equivalence Classes. Modulo Arithmetics.
#17, 9T
Further Set Theory: Modulo Arithmetics continued.
#18, 9Th
Review and Examples
Review and Examples
Review and Examples
Final Exam. TIME TBA 3:00pm - 6:00pm. PLACE TBA (Lecture Room!)