Discrete math is an important branch of mathematics which deals with only discrete (not continuous) values. Discrete objects are usually defined by integers, while continuous objects by real numbers. Discrete math can be explained as a branch of mathematics concerned with the countable sets (A countable set has same cardinality as subsets of natural numbers; it includes rational numbers, but does not include real numbers). In short, the main goal of studying discrete math is the study of discrete objects.
Discrete math is important due to the following reasons:
1) Discrete math plays an important role in college-level and higher mathematics. Discrete calculus and abstract algebra are considered to be the main part of mathematics at college level.
2) Discrete math is more related to real world. Unlike geometry, algebra, trigonometry etc., discrete math enables students to explore interesting real-world applications of mathematics.
3) Discrete math concentrates on proofs and mathematical reasoning. Discrete math promotes creativity and flexibility in thinking out of the box. It would have relatively less number of formulae to memorize.
4) Discrete math is important since it concentrates on fundamental concepts and their application in many different ways.
Main topics in discrete mathematics are:
• Set Theory
Set theory studies sets. The countable sets are dealt under discrete mathematics.
Combinatorics is the study the patterns of combining and arranging discrete structures. Enumerative combinatorics and analytic combinatorics are two kinds of combinatorics. Design theory (combinatorics design study), order theory (study of partially-ordered sets) and partition theory (study of integer partitions) are also considered to be the parts of combinatorics.
• Theoretical Computer Science
It deals with discrete math related to computing. It is mostly based on mathematical logic and graph theory.
• Information Theory
The quantification of information is studied under information theory which is closely connected to coding theory.
The logical formulas are studied under logic or logical mathematics. These are the discrete structures or graph structures.
• Graph Theory
Graph theory deals with networks and graphs. This is the prime subject that is studied in discrete mathematics.
• Discrete Probability Theory
It is a branch of probability that concerns with the events that happen in countable sample spaces.
• Abstract Algebra
Discrete or abstract algebra includes Boolean algebra (programming and logic gates), relational algebra (database), modern algebra (discrete groups, fields and rings and fields).
• Discrete Geometry
Discrete geometry is a branch of geometry that deals with properties of discrete collections of geometrical objects.
The discrete branches of topology include topological graph theory, topological combinatorics, discrete topological space, computational topology and finite topological space.