- Rafia Shabbir

# What is Discrete Mathematics?

Updated: Jan 12, 2021

You may have heard a term discrete mathematics. By its name, you may be thinking that it is probably a branch of mathematics, but it’s not the case. Rather, it is a common name given to the branches of mathematics which have a single common characteristic – i.e. they are discrete instead of continuous. While it may be difficult to define discrete mathematics, but we can say that:

*It is a study of mathematical structures that are distinct and unconnected*

The following mathematical concepts or theories can be put under the banner of discrete mathematics because of their common discrete characteristic:

Graph theory

Logic and Boolean algebra

Number theory

Group theory

Matrix theory

Set theory

Permutations and combinations

Induction and recursion

Partial Order and Lattice

Probability theory

Theory of computation

Algorithmic analysis

**Discrete vs Continuous Sets**

Discrete and continuous sets are opposite to each other.

**Discrete set** in mathematics is defined as a set having unique and distinct elements. Discrete sets can be finite or infinite. For example, consider a set of natural numbers N = {1,2,3,…}. It is discrete because the elements in the set are distinct and there is a strident shift between the elements. In other words, we can say that discrete data take specific values and there are no values in between. The discrete examples from our daily lives involve countable variables such as our age, number of holidays and the number of voters who voted during the elections.

On the other hand, a set of real numbers which includes both rational and irrational numbers is **continuous** because we cannot count the elements in this type of set. For example, all real numbers between 1 and 2 will be written as 1. …, 1.00001… , 1.100001, …..It shows there are infinite real numbers between these two numbers which can neither be counted nor they are discrete. The examples of continuous variables from our daily lives include weight, height, mass, and temperature.

A probability can be discrete or continuous depending on its nature. The most common discrete probability distributions are binomial, Poisson and geometric distribution. On the other hand, normal and continuous probability distributions are examples of continuous probability.

*Interesting fact: Do you know that a digital clock is characterized as having a discrete nature because of the smooth transition of time. On the other hand, an analog clock is termed as continuous because of the separate hour, minute and second hands. There can be infinite different times between let say 1: 10: 20 A.M and 1: 11 A.M.*

**Some Common Uses of Discrete Mathematics**

The comprehension of discrete structures not only assists in a mathematics major, but is also quite worthwhile if you are determined to go for a computer science major. Colleges and universities extend many subjects under a computer science course. These subjects encompass compiler design, databases, operating systems, combinatorics, automata theory, and different programming languages. Discrete mathematics is a foundation of all these computer science courses. The uses of discrete mathematics are widespread, but some of them are listed below:

It helps in the analysis of algorithms

It is helpful in software design specifications and other practical applications

It enhances our problem-solving skills

It encourages analytical thinking and improves mathematical reasoning skills

**Examples of Problems Solved by Discrete Mathematics**

Students often ask how mathematics is applicable in real life. If you are a mathematics teacher, then you might have a hard time answering this question in every class because mathematics courses include abstract concepts like algebra, geometry, and trigonometry. However, while teaching discrete mathematics, you can easily answer this question because the discrete mathematics is helpful in solving problems that students may encounter in their real lives. Some of these problems are listed below:

What is the shortest path from your home to the supermarket?

What is the probability of the occurrence of an event?

Are two computers in the same network connected to each other?

How can you encrypt a message and send it to the person in such a way that no one is able to read the message except the intended person?

How many password combinations are possible given a fixed number of alphanumeric digits?