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Discrete Math

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Discrete Math has a wide application in the modern mathematics, and generally used in decision mathematics. In this, we study about objects, set and graphs. The set of objects studied in discrete mathematics can be finite or infinite. It is an excellent tool for improving reasoning and problem-solving skills, and is appropriate for students at all levels and of all abilities. Before studying discrete mathematics, you must have knowledge of the following parameters.

Matrix: In matrix, we can put any type of object in a rectangular box. These entries can be real or complex. The entries we are doing are called element of matrix.

Permutation and Combination: Permutation of an object is the number of different arrangements that can be made with a given number of set. It is mainly used for small numbers. Whereas combination is the selection of a number of objects taking some or all of them at a time is called Combination. The Order of selection does not matter for the Combination.

Sequence and Series: A given set of number arranged in a definite order or sequence, we can arrange the number in ascending order or descending order.

Series is a mathematical pattern which has some relation with its upcoming member. A Series is the sum of the terms of a Sequence. We can say that, if we have a series like $2,\ 4,\ 8,\ 16$ and so on, then we can say that, $42$ = $82$ = $168$ = $2$. Here, we can see that the common difference is $2$.

Definition

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Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. In simple words, discrete math means the math which deals with countable sets. Discrete math is the study of the data which are not continuous. The topics involved in discrete math 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, rational numbers, automobiles, houses, people etc. So, the main purpose of discrete math is to make the most complex data much easier to analyze and solve. Using 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

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Here we deal with the study of finite or countable data. 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 math 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 a given kind and size.

Set Theory

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Set theory deals with the science of infinite; it is the study of sets which are collection of objects arranged in a group. The language of set theory can be used to define nearly all the mathematical objects. The set of numbers or objects can be denoted by the braces {} symbol.

Axioms of Set Theory:

Set theory has 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:

1) Equality: When two sets have the same elements, then such sets are known to be equal sets. 

2) Empty Set: If a set contains no elements in it, then such sets are called as empty or null sets.

3) Unordered Pair: An unordered pair is the set of theory which contains a set of two elements, $A$ and $B$ which are not related with each other.

4) 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 $A_1$, $A_2$, $A_3$,......, $A_n$ gives a set $A_1$ $\cup$ $A_2$ $\cup$ $A_3$ $\cup$....... $\cup$ $A_n$.

5) 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 finite collections of sets. Say, if we have two sets A and B, then with the help of a Venn diagram, the union and intersection of two set A and B will be shown as below:

Discrete Math Venn Diagram                                Discrete Math Venn Diagrams

Graph theory deals with the study of graphs, which represents the mathematical data diagrammatically so that, we get to know the relationship, differences and variations clearly. The interconnected data represented by mathematical extraction are called as vertices and the links that connect some pairs of vertices called as edges. Graph is a mathematical structure used to pair the relation between objects. Graphs are one of the prime objects of study in Discrete Mathematics.
1) Undirected Graph: Graphs in which edges do not have any orientation or without any ordered pairs are called as undirected graph.

2) Directed Graph: Graphs with n ordered pair is called as directed graph.

3) Mixed Graph: Graphs in which the edges may be both undirected and directed is called as mixed graph.

4) Multigraph: Multigraph is a graph with multiple edges.

5) Quiver: It is a directed graph, which allows loops and multiple arrows between 2 vertices.

6) Simple Graph: This is an undirected graph which has no loops and having only one edge between any 2 different vertices.
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Probability is 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, the greater will be the chances of events to occur. The occurring of the event sometimes can be true and also false. It is just a guide and does not tell us what exactly will happen.
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Number Theory

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Number theory is one of the oldest, largest and fascinating branches 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:

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

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

3) 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. Example: $4 + (-8)$ = $-4$

Integers

4) 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 are not equal to zero. The real number which is not rational is called as irrational number.

For example: $1, 12.38, -0.8625, 34, \sqrt{2}$.

5) 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.

Complex Number