 # Math 55

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

Course Format: Three hours of lecture and two hours of discussion 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.

Textbook: Rosen, Discrete Mathematics and its Applications, 8th 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 2.1-2.5)

Week 3: Division algorithm, modular arithmetic, primes, GCD (Sections 4.1-4.3)

Week 4: Euclidean algorithm, modular exponentiation, solving congruences, Chinese Remainder Theorem, applications to cryptography (Sections 4.4-4.6)

Week 5: Induction and recursion, recursive algorithms (Sections 5.1-5.4; revisit 2.4 for summations)

Week 6: Counting, pigeon hole principle, permutations and combinations, binomial coefficients, distributions, Stirling numbers (Sections 6.1-6.5)

Week 7: Discrete probability theory, conditional probability, independence, random variables (Sections 7.1-7.2)

Week 8: Bayes' Theorem and applications (Section 7.3) Expected value, variance, Chebyshev's inequality (Section 7.4)

Week 9: Recurrence relations and generating functions (Sections 8.1-8.2 and Examples 10-15 in 8.4)

Week 10: Inclusion-exclusion, derangements, formula for Stirling numbers (Sections 8.5, 8.6)

Week 11: Relations, directed graphs, transitive closure, equivalence relations, set partitions, partial orders (Section 9.1, part of 9.3 on digraphs, 9.4-9.6)

Week 12: Graphs, isomorphism, connectivity (Sections 10.1-10.4)

Week 13: Trees, spanning trees, minimum-weight spanning trees (Sections 11.1, 11.4-11.5)

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