## 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 (Sections 1.1-1.6)

**Week 2:** Proof techniques and proof writing (Sections 1.7-1.8)

**Week 3:** Sets, functions, countability and uncountable sets (Sections 2.1-2.5)

**Week 4:** Division algorithm, modular arithmetic, primes, GCD (Sections 4.1-4.3)

**Week 5:** Euclidean algorithm, modular exponentiation, solving congruences, Chinese Remainder Theorem (Sections 4.3-4.4)

**Week 6:** Induction and recursion, recursive algorithms (Sections 5.1-5.4; revisit 2.4 for summations)

**Week 7:** Counting, pigeon hole principle, permutations and combinations, binomial coefficients, distributions, generalized permutations and combinations (Sections 6.1-6.5)

**Week 8:** Discrete probability theory, conditional probability, independence, random variables (Sections 7.1-7.2)

**Week 9:** Bayes' Theorem and applications (Section 7.3) Expected value, variance, Chebyshev's inequality (Section 7.4)

**Week 10:** Recurrence relations (Sections 8.1-8.2), Inclusion-exclusion (Section 8.5)

**Week 11:** Relations, transitive closure, equivalence relations, set partitions, partial orders (Sections 9.1, 9.4-9.6)

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

**Week 13:** Optional topics: Cryptography (Section 4.6), Generating functions (Section 8.4), Applications of inclusion-exclusion (Section 8.6), Trees (Section 11.1), Spanning trees (Section 11.4), More on proof techniques or any of the topics in Weeks 1-12

Note: Optional topics may be incorporated at appropriate points in the semester, at the instructor's choice.

**Documents:**