Discrete Mathematics

Introduction to discrete mathematics pdf:

Discrete mathematics is part of 3 main topics

Mathematics Logic

Boolean Algebra

Graph Theory

discrete mathematics pdf-Mathematics Logic

The find of logic which is used in mathematics is called deductive logic. Mathematical arguments must be strictly deductive in nature. In other words, the truth of the statements to be proved must be established assuming the truth of some other statements.

For example, in geometry we deduce the statement the statement that he sum of the three angles of a triangle is 180 degrees from the statement that an external angle of a triangle is equal to the sum of the other (i.e., opposite) two angles of the triangles of the triangle.

The kind of logic which we shall use here is bi-valued i.e. every statement will have only two possibilities, either True' or 'False' but not both.

Definition:- The symbols, which are used to represent statements, are called statement letters or sentence variables.

To represent statements usually the letters P, Q, R, ..., p, q, r, ... etc., are used

discrete mathematics pdf-Boolean algebra

Boolean algebra was firstly introduced by British Mathematician George Boole (1813 - 1865).the original purpose of this algebra was to simplify logical statements and solve logic problems. In case of Boolean algebra, there are mainly three operations (i) and (ii) or and (iii) not which are denoted by '^^' ,'vv' and (~) respectively. In this chapter, we will use +, . , ' in place of above operations respectively.

Definition:-Let B be a non-empty set with two binary operations + and ., a unary operation ' and two distinct elements 0 and 1. Then B , +, . ,' is called Boolean algebra, if the following axioms are satisfied.

discrete mathematics pdf-Graph theory

Graphs appear in many areas of mathematics, physical, social, computer sciences and in many other areas. Graph theory can be applied to solve any practical problem in electrical network analysis, in circuit layout, in operations research etc.

By a graph, we always mean a linear graph because there is no such thing as a non-linear graph. Thus in our discussion we shall drop the adjective 'linear', and will say simply a 'graph'

Definition:- A graph G = (V, E) consists of a set of objects V = (v1, v2, ...), whose elements are called vertices (or points or nodes) and an another set E = {e1, e2, ....} whose elements are called edges (or lines or branches) such that each ek is identified with an unordered pair (vi, vj) of vertices. The vertices vi and vj associated with the edge ekare said to be the end vertices of ek.

Discrete Mathematics

Introduction to discrete mathematics pdf:

Discrete mathematics is part of 3 main topics

Mathematics Logic

Boolean Algebra

Graph Theory

discrete mathematics pdf-Mathematics Logic

The find of logic which is used in mathematics is called deductive logic. Mathematical arguments must be strictly deductive in nature. In other words, the truth of the statements to be proved must be established assuming the truth of some other statements.

For example, in geometry we deduce the statement the statement that he sum of the three angles of a triangle is 180 degrees from the statement that an external angle of a triangle is equal to the sum of the other (i.e., opposite) two angles of the triangles of the triangle.

The kind of logic which we shall use here is bi-valued i.e. every statement will have only two possibilities, either True' or 'False' but not both.

Definition:- The symbols, which are used to represent statements, are called statement letters or sentence variables.

To represent statements usually the letters P, Q, R, ..., p, q, r, ... etc., are used

discrete mathematics pdf-Boolean algebra

Boolean algebra was firstly introduced by British Mathematician George Boole (1813 - 1865).the original purpose of this algebra was to simplify logical statements and solve logic problems. In case of Boolean algebra, there are mainly three operations (i) and (ii) or and (iii) not which are denoted by '^^' ,'vv' and (~) respectively. In this chapter, we will use +, . , ' in place of above operations respectively.

Definition:-Let B be a non-empty set with two binary operations + and ., a unary operation ' and two distinct elements 0 and 1. Then B , +, . ,' is called Boolean algebra, if the following axioms are satisfied.

discrete mathematics pdf-Graph theory

Graphs appear in many areas of mathematics, physical, social, computer sciences and in many other areas. Graph theory can be applied to solve any practical problem in electrical network analysis, in circuit layout, in operations research etc.

By a graph, we always mean a linear graph because there is no such thing as a non-linear graph. Thus in our discussion we shall drop the adjective 'linear', and will say simply a 'graph'

Definition:- A graph G = (V, E) consists of a set of objects V = (v1, v2, ...), whose elements are called vertices (or points or nodes) and an another set E = {e1, e2, ....} whose elements are called edges (or lines or branches) such that each ek is identified with an unordered pair (vi, vj) of vertices. The vertices vi and vj associated with the edge ekare said to be the end vertices of ek.

Discrete Mathematics

Introduction to discrete mathematics pdf:

Discrete mathematics is part of 3 main topics

Mathematics Logic

Boolean Algebra

Graph Theory

discrete mathematics pdf-Mathematics Logic

The find of logic which is used in mathematics is called deductive logic. Mathematical arguments must be strictly deductive in nature. In other words, the truth of the statements to be proved must be established assuming the truth of some other statements.

For example, in geometry we deduce the statement the statement that he sum of the three angles of a triangle is 180 degrees from the statement that an external angle of a triangle is equal to the sum of the other (i.e., opposite) two angles of the triangles of the triangle.

The kind of logic which we shall use here is bi-valued i.e. every statement will have only two possibilities, either True' or 'False' but not both.

Definition:- The symbols, which are used to represent statements, are called statement letters or sentence variables.

To represent statements usually the letters P, Q, R, ..., p, q, r, ... etc., are used

discrete mathematics pdf-Boolean algebra

Boolean algebra was firstly introduced by British Mathematician George Boole (1813 - 1865).the original purpose of this algebra was to simplify logical statements and solve logic problems. In case of Boolean algebra, there are mainly three operations (i) and (ii) or and (iii) not which are denoted by '^^' ,'vv' and (~) respectively. In this chapter, we will use +, . , ' in place of above operations respectively.

Definition:-Let B be a non-empty set with two binary operations + and ., a unary operation ' and two distinct elements 0 and 1. Then B , +, . ,' is called Boolean algebra, if the following axioms are satisfied.

discrete mathematics pdf-Graph theory

Graphs appear in many areas of mathematics, physical, social, computer sciences and in many other areas. Graph theory can be applied to solve any practical problem in electrical network analysis, in circuit layout, in operations research etc.

By a graph, we always mean a linear graph because there is no such thing as a non-linear graph. Thus in our discussion we shall drop the adjective 'linear', and will say simply a 'graph'

Definition:- A graph G = (V, E) consists of a set of objects V = (v1, v2, ...), whose elements are called vertices (or points or nodes) and an another set E = {e1, e2, ....} whose elements are called edges (or lines or branches) such that each ek is identified with an unordered pair (vi, vj) of vertices. The vertices vi and vj associated with the edge ekare said to be the end vertices of ek.

Discrete Mathematics

Introduction to discrete mathematics pdf:

Discrete mathematics is part of 3 main topics

Mathematics Logic

Boolean Algebra

Graph Theory

discrete mathematics pdf-Mathematics Logic

The find of logic which is used in mathematics is called deductive logic. Mathematical arguments must be strictly deductive in nature. In other words, the truth of the statements to be proved must be established assuming the truth of some other statements.

For example, in geometry we deduce the statement the statement that he sum of the three angles of a triangle is 180 degrees from the statement that an external angle of a triangle is equal to the sum of the other (i.e., opposite) two angles of the triangles of the triangle.

The kind of logic which we shall use here is bi-valued i.e. every statement will have only two possibilities, either True' or 'False' but not both.

Definition:- The symbols, which are used to represent statements, are called statement letters or sentence variables.

To represent statements usually the letters P, Q, R, ..., p, q, r, ... etc., are used

discrete mathematics pdf-Boolean algebra

Boolean algebra was firstly introduced by British Mathematician George Boole (1813 - 1865).the original purpose of this algebra was to simplify logical statements and solve logic problems. In case of Boolean algebra, there are mainly three operations (i) and (ii) or and (iii) not which are denoted by '^^' ,'vv' and (~) respectively. In this chapter, we will use +, . , ' in place of above operations respectively.

Definition:-Let B be a non-empty set with two binary operations + and ., a unary operation ' and two distinct elements 0 and 1. Then B , +, . ,' is called Boolean algebra, if the following axioms are satisfied.

discrete mathematics pdf-Graph theory

Graphs appear in many areas of mathematics, physical, social, computer sciences and in many other areas. Graph theory can be applied to solve any practical problem in electrical network analysis, in circuit layout, in operations research etc.

By a graph, we always mean a linear graph because there is no such thing as a non-linear graph. Thus in our discussion we shall drop the adjective 'linear', and will say simply a 'graph'

Definition:- A graph G = (V, E) consists of a set of objects V = (v1, v2, ...), whose elements are called vertices (or points or nodes) and an another set E = {e1, e2, ....} whose elements are called edges (or lines or branches) such that each ek is identified with an unordered pair (vi, vj) of vertices. The vertices vi and vj associated with the edge ekare said to be the end vertices of ek.

Discrete Mathematics

Introduction to discrete mathematics pdf:

Discrete mathematics is part of 3 main topics

Mathematics Logic

Boolean Algebra

Graph Theory

discrete mathematics pdf-Mathematics Logic

The find of logic which is used in mathematics is called deductive logic. Mathematical arguments must be strictly deductive in nature. In other words, the truth of the statements to be proved must be established assuming the truth of some other statements.

For example, in geometry we deduce the statement the statement that he sum of the three angles of a triangle is 180 degrees from the statement that an external angle of a triangle is equal to the sum of the other (i.e., opposite) two angles of the triangles of the triangle.

The kind of logic which we shall use here is bi-valued i.e. every statement will have only two possibilities, either True' or 'False' but not both.

Definition:- The symbols, which are used to represent statements, are called statement letters or sentence variables.

To represent statements usually the letters P, Q, R, ..., p, q, r, ... etc., are used

discrete mathematics pdf-Boolean algebra

Boolean algebra was firstly introduced by British Mathematician George Boole (1813 - 1865).the original purpose of this algebra was to simplify logical statements and solve logic problems. In case of Boolean algebra, there are mainly three operations (i) and (ii) or and (iii) not which are denoted by '^^' ,'vv' and (~) respectively. In this chapter, we will use +, . , ' in place of above operations respectively.

Definition:-Let B be a non-empty set with two binary operations + and ., a unary operation ' and two distinct elements 0 and 1. Then B , +, . ,' is called Boolean algebra, if the following axioms are satisfied.

discrete mathematics pdf-Graph theory

Graphs appear in many areas of mathematics, physical, social, computer sciences and in many other areas. Graph theory can be applied to solve any practical problem in electrical network analysis, in circuit layout, in operations research etc.

By a graph, we always mean a linear graph because there is no such thing as a non-linear graph. Thus in our discussion we shall drop the adjective 'linear', and will say simply a 'graph'

Definition:- A graph G = (V, E) consists of a set of objects V = (v1, v2, ...), whose elements are called vertices (or points or nodes) and an another set E = {e1, e2, ....} whose elements are called edges (or lines or branches) such that each ek is identified with an unordered pair (vi, vj) of vertices. The vertices vi and vj associated with the edge ekare said to be the end vertices of ek.

Ancient Greek Mathematics

Introduction to ancient greek mathematics :

Ancient greeks are regarded as one of the major discoverer of "Geometry". The greeks were not intrested in numbers much. So they showed their interest in geometry which led to major discoveries. The notable achievements of the greek mathematicians were observed mainly in the period of 6th century BC to 4th century AD.

The word "Mathematics" was termed by pythagoreans (the followers of pythagoras) from the greek word "mathema" meaning " subject of instructions".

Major discoveries in ancient greek mathematics

Some of the major discoveries made by the ancient greek mathematicians are as follows :

The concept of theorems and postulates was introduced by the ancient greek mathematicians.

Euclid's elements were introduced.

One of the most important discovery was theory of conic sections during Hellinistic period.

Archimede's principle was introduced during this period.

Some major contributions were also made in the field of astronomy.

Other achievements were also made in number theory, applied mathematics, mathematical analysis and were close to integral calculus.

Major ancient greek mathematicians

The most famous greek mathematicians are:

Pythagoras

Pythagoras had major contributions in the field of Mathematics. He introduced pythagoras theorem and its proof. He also proved the existence of irrational numbers. He had interests in other fields such as astronomy and philosophy. His studies had a great influence on Plato. He also established an academy with an aim to spread Mathematics in the universe.

Anaxagoras

He was a pre - Socratic greek philospher. He made a significant contribution in the field of cosmology. He studied the celestial bodies closely.

Aristarchus

He was a Greek mathematician and an astronomer. He was the first astronomer to place sun at the center of the solar system instead of Earth. He proposed heliocentric model of solar system. He calculated distance of sun, moon from earth and their sizes.

Thales

He introduced Thales theorem and many corollaries which he used to calculate the height of pyramid and distance of ship from the shore.

Euclid

He introduced a book named Elements. He defined the terms theorems, proofs, postulates etc. His major contribution was conic section.

Archimedes

Archimedes gave an approximate value of Pi. He also calculated area covered by the arc of parabola and had major contribution in area of calculus. He produced the solution for infinite summation series.

Eudoxus

His contributions are observed in modern integration.

and many other greek mathematicians existed during the hellinistic period.

The origins of Greek mathematics are not easily documented. The earliest advanced civilizations in the country of Greece and in Europe were the Minoan and later Mycenean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, they left behind no mathematical documents.

Discrete Mathematics

Introduction to discrete mathematics pdf:

Discrete mathematics is part of 3 main topics

Mathematics Logic

Boolean Algebra

Graph Theory

discrete mathematics pdf-Mathematics Logic

The find of logic which is used in mathematics is called deductive logic. Mathematical arguments must be strictly deductive in nature. In other words, the truth of the statements to be proved must be established assuming the truth of some other statements.

For example, in geometry we deduce the statement the statement that he sum of the three angles of a triangle is 180 degrees from the statement that an external angle of a triangle is equal to the sum of the other (i.e., opposite) two angles of the triangles of the triangle.

The kind of logic which we shall use here is bi-valued i.e. every statement will have only two possibilities, either True' or 'False' but not both.

Definition:- The symbols, which are used to represent statements, are called statement letters or sentence variables.

To represent statements usually the letters P, Q, R, ..., p, q, r, ... etc., are used

discrete mathematics pdf-Boolean algebra

Boolean algebra was firstly introduced by British Mathematician George Boole (1813 - 1865).the original purpose of this algebra was to simplify logical statements and solve logic problems. In case of Boolean algebra, there are mainly three operations (i) and (ii) or and (iii) not which are denoted by '^^' ,'vv' and (~) respectively. In this chapter, we will use +, . , ' in place of above operations respectively.

Definition:-Let B be a non-empty set with two binary operations + and ., a unary operation ' and two distinct elements 0 and 1. Then B , +, . ,' is called Boolean algebra, if the following axioms are satisfied.

discrete mathematics pdf-Graph theory

Graphs appear in many areas of mathematics, physical, social, computer sciences and in many other areas. Graph theory can be applied to solve any practical problem in electrical network analysis, in circuit layout, in operations research etc.

By a graph, we always mean a linear graph because there is no such thing as a non-linear graph. Thus in our discussion we shall drop the adjective 'linear', and will say simply a 'graph'

Definition:- A graph G = (V, E) consists of a set of objects V = (v1, v2, ...), whose elements are called vertices (or points or nodes) and an another set E = {e1, e2, ....} whose elements are called edges (or lines or branches) such that each ek is identified with an unordered pair (vi, vj) of vertices. The vertices vi and vj associated with the edge ekare said to be the end vertices of ek.