Discrete Mathematics

Students will deepen their understanding of logic by exploring Boolean functions and logic circuits, which are fundamental in modern digital systems. They will learn to analyze and simplify Boolean functions, applying these skills to optimize digital circuit design.

This course further develops various number theory concepts encountered in our Methods of Proof course. Students will study different number bases (including binary and hexadecimal) and learn how linear congruences form the backbone of modern cryptographic protocols like Diffie-Hellman and RSA.

Probability is crucial in many real-world applications. This course covers fundamental topics such as elementary probability, random variables, expectation algebra, discrete probability distributions, and combinatorics. Real-world examples are integrated throughout, helping students connect theory to practice.

Students will explore sequences and series, covering topics such as finite sums, geometric series, and the generalized binomial and multinomial theorems. Students will also study first and second-order recurrence relations and learn how generating functions can be used to find solutions to these relations. Graphs are a fundamental part of modern computing systems. Students will explore graph theory, prove key properties of graphs, and study essential graph traversal algorithms, including breadth-first and depth-first search, spanning tree algorithms, and Dijkstra's algorithm. Finally, students learn about the theory underpinning modern computer algorithms. They will study theoretical mathematical machines (acceptors, transducers, and Turing machines) and asymptotic notation to describe an algorithm's complexity.

Boolean Algebra

• Understanding the concept of a Boolean function, including their representations, composition, equivalence, and counting the number of Boolean functions on N variables.

• Understanding the definitions of exclusive disjunction (XOR), Peirce arrow (NOR), Sheffer Stroke (NAND), and simplifying Boolean expressions involving these operations.

• Understanding disjunctive and conjunctive normal forms and expressing Boolean functions in principle disjunctive and conjunctive normal forms.

• Working with Boolean polynomials, including simplifying them and proving various properties related to these polynomials.

• Using Karnaugh maps to express a Boolean function in minimal disjunctive normal form.

• Understanding the concepts of truth-preservation, self-duality, monotonicity, and affineness in relation to Boolean functions.

• Applying Post's theorem to determine whether a set of Boolean functions is functionally complete.

• Representing a logic circuit using Boolean functions and simplifying logic circuits.

Number Theory

• The binary system of numbers, including converting between binary and decimal representations of integers, adding, subtracting, multiplying, and dividing binary integers, and representations of binary fractions.

• Systems of numbers in arbitrary bases, including hexadecimal numbers.

• Understanding and applying Fermat's little theorem.

• Working with Euler's totient function and its properties and applying Euler's theorem.

• Cryptography fundamentals, including the Caesar and linear ciphers, the Diffie-Hellman protocol, and the RSA cryptosystem.

Probability & Combinatorics

• Understanding and applying the law of total probability and Bayes' Theorem.

• Working with probability mass functions and cumulative distribution functions for discrete random variables.

• Calculating expected values, variance, and moments for discrete random variables.

• Understanding and applying properties of expectation and variance.

• Constructing and applying the probability mass function (PMF) and cumulative distribution function (CDF) of well-known probability distributions, including:

  • The discrete uniform distribution

  • The Bernoulli distribution

  • The binomial distribution

  • The Poisson distribution

  • The geometric distribution

  • The negative binomial distribution

  • The hypergeometric distribution

• Understanding the relationship between the Bernoulli and binomial distributions and the geometric and negative binomial distributions.

• Solving real-world situations using these distributions and understanding the conditions where modeling with a known distribution is appropriate.

• Approximating a binomial random variable using a Poisson distribution.

• Calculating permutations and combinations, including cases with repetition, and modeling real-world and mathematical problems using combinatorial techniques.

• Understanding and applying the inclusion-exclusion and pigeonhole principles and constructing partitions of natural numbers.

Sequences

• Calculating sums of finite linear, quadratic, and cubic series.

• Working with infinite geometric series and testing for convergence.

• Working with the generalized binomial theorem, including calculating the range of validity of a given expansion and using a binomial expansion to estimate a given number.

• Applying the multinomial theorem and counting the number of terms in a multinomial expansion.

Recursion

• Understanding the concept of a recurrence relation and fully classifying a given relation.

• Solving first and second-order homogeneous recurrence relations using the characteristic equation method.

• Solving first and second-order inhomogeneous recurrence relations using the unknown coefficients method.

• Understanding the concept of a generating function and using them to solve homogeneous recurrence relations.

• Working with generating functions of inhomogeneous recurrence relations.

Graph Theory

• Understanding the concept of a graph and basic graph terminology, including graph isomorphism, connectivity, subgraphs, walks, paths, trails, and distances in undirected and directed graphs.

• Understanding and applying the degree sum formula and proving the handshaking lemma.

• Working with complete, bipartite, planar, and regular graphs.

• Trees, spanning trees, and finding minimum spanning trees by inspection.

• Counting the number of labeled trees, rooted trees, and rooted forests on N vertices.

• Representing graphs using incidence, adjacency, and distance matrices.

• Understanding the concepts of directed acyclic graphs (DAG) and topological orderings.

• Applying Kahn's algorithm to find a topological ordering of a DAG.

• Applying the graph-traversal algorithms of the breadth-first search (BFS) and depth-first search (DFS).

• Kruskal's and Prim's algorithms are applied to find minimum spanning trees, including Prim's algorithm on a distance matrix.

• Applying BFS to find shortest distances in unweighted graphs and Dijkstra's algorithm in weighted graphs.

The Theory of Algorithms

• Understanding the difference between finite-state transducers and acceptors.

• Representing Mealy and Moore machines using various methods and processing strings using these machines.

• Understanding the difference between deterministic and nondeterministic finite automata, representing them in various forms, and processing strings using these machines.

• Describing the language of simple automata and regular expressions.

• Representing Turing machines using various methods and processing strings using these machines.

• Understanding and applying the definition of Big-O, Big-Omega, and Big-Theta in the context of algorithmic complexity.