Discrete

Discrete, in general use, means separate or distinct. In mathematics, the term denotes objects that can be counted and are not continuous. Examples of discrete sets include the integers, finite collections, and graphs; by contrast, real-valued quantities on an interval form a continuous set.

Discrete mathematics is the study of such discrete structures and their properties. It includes combinatorics, graph theory, number theory, logic, and set theory, as well as discrete probability, algorithms, and formal languages. The subject emphasizes finite or countable objects and often focuses on constructive methods, algorithms, and complexity.

Applications span computer science, cryptography, networking, scheduling, coding theory, and software verification. In statistics and data analysis, discrete data take distinct values (for example counts or categories) rather than any value within a range. In signal processing and computer engineering, discrete-time signals and digital representations arise from sampling and quantization of continuous phenomena.

Historically, discrete mathematics developed alongside the rise of computation and information theory in the 20th century. It provides models and tools for reasoning about systems that evolve in steps or that can be enumerated, making it essential to algorithms, data structures, and formal reasoning about software and networks.