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Discrete Mathematics
Engineering & Social Sciences Program
Madrid, Spain
Dates: early Sep 2025 - mid Dec 2025
Discrete Mathematics
OVERVIEW
CEA CAPA Partner Institution: Universidad Carlos III de Madrid
Location: Madrid, Spain
Primary Subject Area: Mathematics
Instruction in: English
Course Code: 16489
Transcript Source: Partner Institution
Course Details: Level 200
Recommended Semester Credits: 3
Contact Hours: 42
Prerequisites: Calculus I and II, and Linear Algebra
DESCRIPTION
1. Arithmetic
1.1 Integers
1.2 Division algorithm
1.3 Largest common divisor: Euclid's algorithm
1.4 Prime numbers and the Fundamental Theorem of Arithmetic
1.5 Diophantine equations
1.6 Congruences: modular arithmetic
2. Elementary set theory
2.1 Basic notions
2.2 Set operations and properties
2.3 Functions
2.4 Relations: equivalence and order
2.5 Cardinality
3. Combinatorics
3.1 Elementary counting rules: sum and product
3.2 Pigeon-hole principle
3.3 Permutations and combinations
3.4 Binomial coefficients
3.5 Principle of inclusion and exclusion
3.6 Derangements
3.7 Generating functions
3.8 Partitions
3.9 Recurrences
4. Introduction to groups
4.1 Law of composition
4.2 Groups and subgroups
4.3 Homomorphisms and isomorphisms
4.4 Cyclic groups
4.5 Cosets, Lagrange's theorem, and quotient groups
4.6 Applications to cryptography
5. Fundamentals of graph theory
5.1 Definitions and examples
5.2 Matrix representations
5.3 Eulerian and Hamiltonian graphs
5.4 Trees
5.5 Optimisation and matching
5.6 Planar graphs
5.7 Directed graphs
5.8 Networks
1.1 Integers
1.2 Division algorithm
1.3 Largest common divisor: Euclid's algorithm
1.4 Prime numbers and the Fundamental Theorem of Arithmetic
1.5 Diophantine equations
1.6 Congruences: modular arithmetic
2. Elementary set theory
2.1 Basic notions
2.2 Set operations and properties
2.3 Functions
2.4 Relations: equivalence and order
2.5 Cardinality
3. Combinatorics
3.1 Elementary counting rules: sum and product
3.2 Pigeon-hole principle
3.3 Permutations and combinations
3.4 Binomial coefficients
3.5 Principle of inclusion and exclusion
3.6 Derangements
3.7 Generating functions
3.8 Partitions
3.9 Recurrences
4. Introduction to groups
4.1 Law of composition
4.2 Groups and subgroups
4.3 Homomorphisms and isomorphisms
4.4 Cyclic groups
4.5 Cosets, Lagrange's theorem, and quotient groups
4.6 Applications to cryptography
5. Fundamentals of graph theory
5.1 Definitions and examples
5.2 Matrix representations
5.3 Eulerian and Hamiltonian graphs
5.4 Trees
5.5 Optimisation and matching
5.6 Planar graphs
5.7 Directed graphs
5.8 Networks