What is Discrete Mathematics? 

The word ‘discrete’ means individual or separate. In mathematics, discrete mathematics is a study of discrete elements that involves algebra and arithmetic. Discrete mathematics has so many applications in computer science and practical mathematics. It also helps us in improving our reasoning and problem-solving skills. 

In the ancient times, people used to send messages from one country to other country. These messages were sent using letters. Sometimes, the information about wars were also sent to kings. But this information had to be kept safe. If the secret messages are leaked, then it is dangerous because the opponents will come to know the idea and they may even change the contents of the message. So, the message has to safely reach. This can be done if the message is conveyed through codes. If the language is not explicit, the reader might not understand the idea at all. These codes are framed using some rules. For example, the number 1 represents the letter ‘a’ and 2 represents the letter ‘b’. Guess what the number 15 represents? In this way any message can be encoded. To read the message, the reader needs to have the decoding key. This will prevent the message from corruption from any other external source and the message is also preserved. This study is called as cryptography and it is a branch of discrete mathematics. Even in the modern times, securing social media messages is done using cryptography. The message will be end-to-end encrypted and no one else can read the message by this technique. 

• Google maps use the concept of discrete mathematics to tell us the fastest driving route and time to reach our destination. 
• Connecting networks like social media or roads are all some examples of discrete structures. 
• There are different ways in which voting can be done. The information about the number of people to vote and some other relevant information help to choose which method. This is done by discrete mathematics. 

Classification of Mathematics

Mathematics is grouped into two categories:

Continuous Mathematics

Continuous mathematics takes two points in the real line. The two points are two real numbers and the variables can take any value within this range.

Example:

• A baby’s height could be any value within the range of baby height chart which is not just certain fixed height.
• The animal height.
• Time in marathon race.
• Length of branches in a tree.

Here the numbers are not fixed values. That is, they either keep increasing or decreasing smoothly without any break. This is called Continuous mathematics.

Discrete Mathematics

Discrete mathematics considers only fixed values.

Example:

• Number of students in a school, here the count will be fixed and there can never be half a student.
• The sum of outcomes in rolling a pair of dice, which can only be 2,3,4,5,6,7,8,9,10,11 and 12.
• Boolean algebra involves logic. Here, the numbers 0 and 1 are used to denote false or true. Consider the statement ‘The leaves are green in color’. The statement is either false or true. 

Classification of Discrete Mathematics

Combinatorics 

Combinatorics is the probable ways of selecting some object out of a collection and its number of ways of arrangement. They are calculated using the combinations and permutations. When order doesn’t matter to choose objects from a group, it is called a combination. When the order does matters, it is called permutation.

A permutation is calculated by arranging R distinct objects out of collection of N objects. The formula for a permutation is:

A combination is calculated by choosing R distinct objects out of collection of N objects. The formula for a combination is:

Graph Theory 

Graph theory is a field in discrete mathematics. It is a study about graphs. A graph is represented by points called as nodes and lines called as edges. Graph is made of non-empty set of nodes and edges. Graphs help in easily visualizing any complicated problem. Graph theory is mainly used for the simplification of works like construction of roads, bridges and for the transportation purposes. Usage of algorithms makes this easy. Suppose the problem is to draw a graph to connect two cities using lines. What is the route that is of minimum distance between the two cities so that the time taken for transportation process is quick? This problem-solving is done easily in graph theory. 

Number Theory  

Number theory is a branch of mathematics which deals with discrete sets like Natural numbers and integers.  

Logic 

Logic is one of the branches of discrete mathematics. What is logic? Suppose it is suggested that -1 is the smallest natural number. Agreeing to it or not becomes a question. Definitely there would be disagreement as there is no logic behind this statement. It is because -1 is not a natural number. Therefore, it cannot be the smallest natural number too. Logical things are only believed. Constructing a proof is necessary to demonstrate that the statement is valid.  

Suppose that someone says the weather is cool in winter season because the earth is facing away from the sun. Then the argument is valid because it is scientifically proven. Thus, logic helps in verifying a statement. 

Conjunction (AND)

• The conjunction statement P˄Q has the value T only when both P and Q have the truth value T.
• The conjunction statement P˄Q has the value F when either P or Q or both have the truth value F.

Disjunction (OR)

• The disjunction (OR) statement P˅Q has the truth value T when either P or Q or both have the truth value T.
• The disjunction (OR) statement P˅Q has the truth value F when both P and Q have the truth value F.

Set Theory

A group of defined and distinguishable collection of objects is called a set. Set theory is a study about the collection of well-defined elements termed as set.

Example,

• Set of all states in a country.
• Set of all positive integers.
• Set of lowercase letters in English alphabet

Context and Applications 

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for 

• B.Sc in Maths
• M.Sc in Maths