Discrete Mathematics

Discrete mathematics is the branch of mathematics dealing with objects that consider only distinct, separated values. This concept is in contrast to continuous mathematics, which deals with objects that can vary smoothly and continuously.

Sets

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Operations such as unions, intersections, and complements can be performed on sets.

For example, if we have a set A = {1, 2, 3} and set B = {2, 3, 4}, the union of A and B is A ∪ B = {1, 2, 3, 4} and the intersection of A and B is A ∩ B = {2, 3}.

Logic

Logic is the study of formal reasoning based on statements or propositions. Propositional logic deals with propositions and logical connectives ('and', 'or', 'not'), while predicate logic includes variables and quantifiers ('for all', 'there exists').

For example, given two statements P: "It is raining" and Q: "I will stay indoors", we could form the compound statement "If it is raining, then I will stay indoors", represented as P ⇒ Q in propositional logic.

Combinatorics

Combinatorics is the study of counting, arrangement, and combination. It includes concepts like permutations (arrangements of objects in a certain order) and combinations (selection of objects without regard for the order).

For example, if we have 5 books and we want to know in how many ways we can arrange 3 of them on a shelf, we use the concept of permutation. The answer would be P(5, 3) = 5! / (5-3)! = 60 ways.

Graph Theory

Graph theory studies graphs, which are mathematical structures used to model pairwise relations between objects. A graph is made up of vertices (also called nodes or points) which are connected by edges (also called arcs or lines).

For example, a social network can be represented as a graph, where individuals are nodes and the relationship between them are edges.

Applications

Discrete mathematics is foundational material for computer science: for example, computers have built-in capabilities of manipulating and making decisions based on discrete data. It's also used in graph theory, logistics, statistics, data science, artificial intelligence, algorithm design, and more.