Teacher: Anastasia Trofimova
The course topics follow those of the Data Science and Business Analytics’ basic course but will also prove useful for Applied Mathematics and Informatics and Software Engineering students as well. The course is student oriented. It includes many problems from the very simple ones to the most intriguing.
Course Plan :
1. Logic. Statement, predicates and quantifiers. Tautology. Proofs by definition and by contraction.
2. Set theory. Operations with sets. Cardinality. Properties of operations with cardinalities.
Countable and Uncountable Sets. Mappings. The inclusion–exclusion principle for the cardinality of sets.
3. Type of proofs: mathematical induction, the least number principle, recursion, proof by counterexample, existence proofs.
4. Divisibility and modular arithmetic, great common divisor, analysis of remainders, well-ordering.
5. Diophantine equations. Fundamental theorem of arithmetic. The Chinese remainder theorem. Euclid's algorithm. Fermat’s Little Theorem. Euler theorem.
6. Relations and functions. Injections, surjections, bijections. Composition. Inverse functions. Relations of equivalence and order.
7. Equivalent sets. Countable sets. Proofs of equivalence. Cantor-Schroder-Bernstein theorem.
6. Combinatorics. Permutations. Arrangements. Combinations. Binomial coefficients. The stars and bars method. The Pigeonhole Principle.
7. Binary relations. Strict and non-strict partial orders. Upper and lower bounds. Linear orders, chains and antichins. The equivalence relations.
8. An Introduction to Probability Theory. Sample Space, Outcomes, Events, Probability. Random Variables and their Distributions. Conditional Probability and Independence. Expectation of a Random Variable. Variance, Standard Deviation, Chebyshev’s Inequality. Law of Large Numbers. Central Limit Theorem.
9. Graphs. Types of graphs and their applications. Handshaking lemma. Cycles. Spanning thee. Colorings.
These topics are preliminary. Students who will enroll for the course can ask to add new topics.
Course structure and Grading: 1 class per week from the end of September(10-12 classes in first semester, 5-6 in third module), 7-8 homeworks.
For DSBA freshmen.
Timetable: Thursday 6.10pm from September, 30