Abstract Algebra

Sets

Disjoint Set -Two sets are disjoint if they have no elements in common Example X= {1,2,3} and Y= {4,5} X∩Y=ф

-is a set whose elements are in the set A or in set B Union Set -is a set whose elements are in the set A or in set B Example A= {3,6,9,12} and B= {0,6,12,18} AUB= {0,3,6,9,12,18}

-is a set whose elements are in both A and B Intersection Set -is a set whose elements are in both A and B Example A= {3,6,9,12} and B= {0,6,12,18} A∩B= {6,12}

-is a set whose element/s are not in the universal set Complement Set -is a set whose element/s are not in the universal set Example U= {1,2,3,4,5,6,7,8,9} A= {1,2,3,4,5,6} A1= {7,8,9}

-is a set that has limited number of elements Finite Set -is a set that has limited number of elements Example U= {Rainbow Colors}

-is a set that has unlimited number of elements Infinite Set -is a set that has unlimited number of elements Example U= {All counting numbers}

Equivalent Set -is a set with the same cardinal number but the elements need not to be the same Example A= {a,b,c,d} and B= {k,l,m,n} A=B

-is a set with exactly the same elements Equal Set -is a set with exactly the same elements Example A= {a,b,c,d} and B= {b,c,a,d} A~B

Permutations

Combinations

Evaluating Imaginary Numbers

Evaluating Imaginary Numbers

Sample Spaces represents the total number of sample points Sample Spaces represents the total number of sample points. Sample Space is determined as permutation and combination

Famous Mathematicians

Nikolay Ivanovich Lobachevsky

A Russian mathematician who was one of the first to apply a critical treatment to the fundamental postulates of Euclidean Geometry. He devised the method of non Euclidian Geometry independently from the hungarian mathematician Janos Bolyai and Gauss. His Writings include: “Principles of Geometry” (1829), “Imaginary Geometry” (1835) and “New Principles of Geometry with a complete Theory of Parallels” (1835).

Gaspard Monge

Recognized as the inventor of Descriptive Geometry Recognized as the inventor of Descriptive Geometry. Monge’s general theory of curvative of geometric surfaces established the basis for much of the subsequent work of the German Mathematician Carl Friedrich Gauss in this field.

Alfred North Whitehead

Generally recognized as one of the greatest 20th century philosophers. A brilliant mathematician who made lasting contributions in the field of mathematics.

Whitehead has deep knowledge in Philosophy and Literature and this lead him to the study of the foundations of Mathematics, Philosophy of Science and development of Symbolic Logic. He wrote “A Treatise on Universal Algebra” (1898), “The Principle of Relativity” (1922) and “An Introduction to Mathematics” (1922) for the general reader.

Karl Theodor Wilhelm Weierstrass

A German Mathematician who helped found the modern theory of Functions A German Mathematician who helped found the modern theory of Functions. He was known as the Father of Modern Analysis. He laid the foundations for the arithmetization of mathematical analysis through rigorous development of the real number system.

Norbert Wiener

The founder of Cybernetics, the study of control and communication in machines, animals and organization. He wrote “Cybernetics” (1948), “The human use of Human Beings” (1950), “Nonlinear Problems of Random Theory” (1958), “The Temper” (1959) and “God and Golem Inc.” (1964)

Georg Cantor

His early work with the Fourier Series led to his development of the theory of Irrational sets, upon which modern mathematical analysis is based. This theory extended the concept of numbers by introducing the infinite, or as he called them: transfinite numbers. His work was largely responsible for the foundations of Mathematics and Mathematical Logic.

Gottlob Frege

The founder of Modern Mathematical Logic The founder of Modern Mathematical Logic. He invented many symbolic notations such as quantifiers and variables. He published: “Conceptual Notation” (1879), “The Foundation of Arithmetic” (1884) and “The Basic Laws of Arithmetic: An Exposition of the System” (1965).

Florian Cajori

His works on the History of Mathematics included: “The Teaching and History of Mathematics in the U.S.” (1890), “A History of Mathematics” (1894) and “A History of Mathematical Notations” (1928-1929).

Karl Pearson

Known for the development of some central techniques of Modern Statistics and for applying these techniques to the problem of biological inheritance. His research led to defining the meanings of correlation, regression analysis and standard deviation. He published the book entitled: “ The Grammar of Science” (1892).

Charles Babbage

He designed and built mechanical computing machines on principles that that anticipated the modern electronic computer. In the 1820’s he began developing his Difference Engine, a mathematical device that can perform simple mathematical calculations. Babbage’s book “Economy of Machines and Manufactures” initiated the field of study known today as Operational Research.