# Math 55

### Math 55 - Discrete Mathematics -- [4 units]

**Course Format: **Three hours of lecture and two hours of
discussion/workshop per week; at the discretion of the instructor, an
additional hour of discussion/workshop or computer laboratory per week.

**Prerequisites: **Mathematical maturity appropriate to a sophomore math class. 1A-1B recommended.

**Credit Option: **Students will receive no credit for 55 after taking Computer Science 70.

**Description: **Logic, mathematical induction sets, relations, and
functions. Introduction to graphs, elementary number theory,
combinatorics, algebraic structures, discrete probability theory.
(F,SP)

**Textbook: ** Kenneth H. Rosen, **Discrete Mathematics and Its Applications**, McGraw Hill, custom UC Berkeley paperback edition.

**Outline of the Course:**

**Week 1:** Propositional logic, quantifiers, rules of inference, proof techniques (Chapter 1)

**Week 2:** Sets, functions, countability and uncountable sets
(Sections 2.1-2.3, and part of 2.4 on cardinality) Algorithms, halting
problem (Section 3.1), also discuss undecidability in lectures

**Week 3:** Division algorithm, modular arithmetic, primes, GCD (Sections 3.4-3.5)

**Week 4:** Euclidean algorithm, modular exponentiation, solving
conguences, Chinese Remainder Theorem, applications to cryptography
(Sections 3.6-3.7)

**Week 5:** Induction and recursion, recursive algorithms (Sections 4.1-4.4; revisit 2.4 for summations)

**Week 6:** Counting, pigeon hole principle, permutations and
combinations, binomial coefficients, distributions, Stirling numbers
(Sections 5.1-5.5)

**Week 7:** Discrete probability theory, conditional probability, independence, random variables (Sections 6.1-6.2)

**Week 8:** Bayes' Theorem and applications (Section 6.3) Expected value, variance, Chebyshev's inequality (Section 6.4)

**Week 9:** Recurrence relations and generating functions (Sections 7.1-7.2 and Examples 10-15 in 7.4)

**Week 10:** Inclusion-exclusion, derangements, formula for Stirling numbers (Sections 7.5, 7.6)

**Week 11:** Relations, directed graphs, transitive closure,
equivalence relations, set partitions, partial orders (Section 8.1, part
of 8.3 on digraphs, 8.4-8.6)

**Week 12:** Graphs, isomorphism, connectivity (Sections 9.1-9.4)

**Week 13:** Trees, spanning trees, minimum-weight spanning trees (Sections 10.1, 10.4-10.5)

**Week 14 and 15:** At the instructor’s discretion

Additional topics of the instructor's choice to be incorporated at appropriate points to complete the full semester.

**Other Material:** The instructor may cover further topics (e.g.,
advanced counting techniques, error-correcting codes, graphs and trees,
binary relations).