Discrete Mathematics

The study of collections of objects called sets, including operations such as union, intersection, difference, and complement. Set theory provides the foundational language for nearly all of modern mathematics and is essential for defining functions, relations, and data structures.

Propositional logic studies statements that are either true or false and the connectives (AND, OR, NOT, implication) that combine them. Predicate logic extends this with quantifiers (for all, there exists) and predicates that describe properties of objects, enabling more expressive reasoning.

The study of graphs, which are mathematical structures consisting of vertices (nodes) connected by edges (links). Graph theory models pairwise relationships and is used to analyze networks, optimize routes, schedule tasks, and solve connectivity problems.

The branch of mathematics concerned with counting, arrangement, and combination of objects according to specified rules. It includes permutations, combinations, the pigeonhole principle, and generating functions, and is critical for analyzing algorithm complexity.

A proof technique used to establish that a statement holds for all natural numbers. It consists of proving a base case and then showing that if the statement holds for an arbitrary case $n$, it also holds for $n + 1$ (the inductive step).

A relation is a set of ordered pairs that defines a relationship between elements of two sets. Functions are special relations where each input maps to exactly one output. Properties of relations include reflexivity, symmetry, transitivity, and equivalence.

Equations that define a sequence recursively, where each term is expressed as a function of preceding terms. They are widely used to analyze the running time of recursive algorithms and to model growth processes.

An algebraic structure that captures the essential properties of logical operations and set operations. It operates on binary values (true/false or 1/0) with AND, OR, and NOT operations, and is fundamental to digital circuit design and computer architecture.

A tree is a connected acyclic graph. Trees model hierarchical structures such as file systems, organizational charts, and decision processes. A spanning tree of a graph is a subgraph that includes all vertices and is a tree, useful for network design and optimization.

A system of arithmetic for integers where numbers 'wrap around' after reaching a certain value called the modulus. It is fundamental to number theory, cryptography, hashing algorithms, and error-detecting codes.