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# Discrete Math

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 Sub Topics Discrete Math has a wide application in the modern mathematics. Discrete mathematics is generally used in decision mathematics. In this, we study about objects, set and graphs. Before studying discrete mathematics, you must have knowledge of the following parameters. Matrix: In matrix, we can put any type of object in the rectangular box. These entries can be real or complex. The entries we are doing are called element of matrix. In matrix, we are having rows and columns as well. Permutation: Permutation of an object is the number of different way it can be ordered. It is mainly used for small numbers. For example, if we have to choose 6 coins out of 9, then we will use permutation for that. It is mainly of two types: Permutation with repetitionPermutation without repetition Combination: If we have to choose a large number of object, your way of choosing is also important. Like permutation, combination is also of two types: Combination with repetitionCombination without repetition Sequence: If we have a given set of number and we arranged the numbers in a sequence, we can arrange the number in ascending order or descending order. Series: Series is a mathematical pattern which has some relation with its upcoming member. We can say that, if we have a series like 2, 4, 8, 16 and so on, then we can say that, $\frac{4}{2}$ = $\frac{8}{2}$ = $\frac{16}{8}$ = 2. Here, we can see that the common difference is 2.

## Discrete Math Definition

In simple words, discrete math means the maths which deals with countable sets. Discrete math is the study of the data which are not continuous. The topics involved in discrete maths are:
• Integers
• Graphs
• Statements in logic
Discrete mathematics deals with the discrete objects. Discrete objects are the one which can be separated from each other and can be countable. For example, integers, automobiles, houses, people etc.. So, the main purpose of discrete maths is to make the most complex data much easier to analyze and solve them. Using the discrete math, we can handle infinity, large quantity data very easily and the results from the formal approach can be reusable.

Therefore, discrete mathematics can be defined as the study of mathematical structures that are fundamentally discrete rather than continuous.

## Combinatorics

Combinatorics is a branch of mathematics dealing with the study of finite or countable datas. Combinatorics involves the mathematical relations that characterize their properties, like combination and permutation of the sets, enumeration etc. Sometimes, the combinatorics also refers to a larger subset of discrete maths which include graph theory.

The main purpose of the combinatorics is to decide when a certain criteria should be met, constructed and analyzed and conclude whether they are the largest, smallest or optimal objects for the data of given kind and size.

## Set Theory

Set theory is one of the important concept of mathematics which deals with the science of infinite. In other words, set theory is the branch of mathematics that deals with the study of sets, collection of objects. The language of set theory can be used to define nearly all the mathematical objects.

Axioms of Set Theory:

Set theory have some axioms which are used in mathematics. The theorems and rules are based on these axioms. And, these axioms are purely accepted on faith.

Following are some of the axioms:
• Equality: When two sets have the same elements, then such sets are known to be equal sets
• Empty set: If a set contains no elements in it, then such sets are called as empty or null sets.
• Unordered Pair: An unordered pair are the set of theory which contains a set of two elements, a and b, which are not related with each other.
• Union of sets: The union of sets, denoted as $\cup$ is the set of all the distinct elements in the collection. For example, the union of the sets A1, A2, A3, ......, An gives a set A1 $\cup$ A2$\cup$ A3 $\cup$ ........ $\cup$ An.
• Power of sets: The set of all the subsets of set 'A' is called the power set of A and it is denoted as P(A).

Venn Diagram:

Venn diagram is a schematic diagram which is commonly used to represent the set theory. It helps us to know all the possible logical relationship between a finite collection of sets. Say, if we have two sets A and B, then with the help of Venn diagram, the union and intersection of two set A and B will shown as below:

## Graph Theory

Graph theory is the branch of mathematics which deals with the study of graphs, which represents the mathematical data diagrammatically so that, we get to know the relationship, differences, variations clearly. The interconnected data represented by mathematical extraction are called as vertices and the links that connect some pairs of vertices are called as edges. Graphs are one of the main object in discrete mathematics. There are different types of graphs. Given below are few of them:
• Undirected graph: A graph in which edges do not have any orientation or without any ordered pairs are called as undirected graph.
• Directed graph: A graph with n ordered pair is called as directed graph.
• Mixed graph: A graph in which the edges may be both undirected and directed is called as mixed graph.
• Multigraph: Generally, multigraph is a graph with multiple edges.
• Quiver: It is a directed graph, which allows loops and multiple arrows between 2 vertices.
• Simple graph: This is an undirected graph which has no loops and having only one edge between any 2 different vertices.

## Probability

Probability is a branch of mathematics that deals with the study of certainty. It is just an estimated or guessed statement of something that is yet to occur. The higher the degree of probability, greater the chances of events to occur. The occurring of the event sometimes can be true and also false. It is just a guide. It does not tell us what exactly will happen.

## Number Theory

Number theory is one of the oldest, largest and a fascinating branch of mathematics. It mainly deals with the properties of the numbers and its functions. A mathematical object which is used to count, label and measure is called as number. Number set is a foundation for all the mathematical operations like addition, subtraction, multiplication and division. The study of these numerical operations is called as Arithmetic. The number theory classifies the numbers into many types as follows:

Natural Number:
Natural number are counting number. The sequence of natural number starts with 1. The counting numbers are (1, 2, 3, 4, 5 .......).

Whole Number:
The set of numbers which starts from 0 to infinity is called as whole number. For example: (0, 1, 2, 3, 4, 5, ............).

Integers:
Integers include both positive and negative whole numbers. These can be written without a fractional or decimal component. Whole number greater than zero is called positive integers and whole number less than zero are called as negative integers. Zero is neutral.

Rational Number:
The number which can be expressed in fraction is called as rational number. Fractions are normally expressed with two integers and denominator which is not equal to zero. The real number which is not rational is called as irrational number.
For example: 1, 12.38, -0.8625, $\frac{3}{4}$, $\sqrt{2}$.

Complex Number:
Complex number is the combination of real number and an imaginary number. The complex number can be expressed in the form of "a + bi", where, 'a' and 'b' are real numbers and 'i' is an imaginary number. Imaginary number allows us to take the square root of negative number.