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Introduction to Discrete Mathematics

Discrete mathematics is the study of mathematical structures which are discrete instead of continuous. It is different from the real numbers which have the property of varying "smoothly", the objects which are studied in discrete mathematics - such as graphs, integers, and statements in logic - do not differ smoothly as real number, but have distinct, separated values. And hence, discrete mathematics does not include the topics in "continuous mathematics" such as analysis and calculus. Discrete objects can frequently be enumerated by integers. More particularly, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (sets that have similar cardinality as subsets of the natural numbers that is involving rational numbers but not real numbers). Though, there is no definition of the term "discrete mathematics" that is universally agreed and exact. In fact, discrete mathematics is illustrated less by what is included than by what is excluded: continuously varying the quantities and related ideas.

The set of objects studied in discrete mathematics can be infinite or finite. The term finite mathematics is sometimes applied to sections of the field of discrete mathematics which deals with finite sets, mainly those regions which are relevant to the business.

Research in the field of discrete mathematics increased in the latter half of the 20th century partially as of the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts from discrete mathematics are helpful in studying and describing objects and problems in branches of computer science, such as computer algorithms, automated theorem proving, cryptography, programming languages, and software development. On the contrary computer implementations are important in applying ideas from discrete mathematics to real-world problems, such as in the operations research.

However the major objects of study in discrete mathematics are discrete objects, analytic techniques from continuous mathematics are frequently employed also.

**Graph**:

What are Graphs?

- Common meaning in everyday math:

A plot or a chart of numerical data using a coordinate system.

- Scientific meaning in discrete mathematics:

A specific class of discrete structures which is useful for representing relations and has a convenient webby-looking graphical representation.

**Tree**:

- Define: - A tree is a connected undirected graph with no simple circuits.
- As tree cannot have a circuit, a tree cannot consist of multiple edges or loops.
- Therefore, any tree should be a simple graph.
- Theorem:

- Any undirected graph is a tree if and only if there is a exclusive simple path between any two of its vertices.

- In common, we use trees to represent the hierarchical structures

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