Logic Of George Boole And Its Application To The Design Of Modern Computers

Modern computers are now considered as one of the most important digital machines in our modern time. People use computers in many ways. In business, computers track inventories with bar codes and scanners, check the credit status of customers, and transfer funds electronically. Computers in automobiles regulate the flow of fuel, thereby increasing gas mileage. Computers also entertain, creating digitized sound on stereo systems or computer-animated features from a digitally encoded laser disc. Computer programs, or applications, exist to aid every level of education, from programs that teach simple addition or sentence construction to programs that teach advanced calculus. But did you know that that the design of modern computers originally came from the logician George Boole who developed the Boolean algebra?

Boolean algebra is a two-valued system of algebra that represented logical relationships and operations. Later, scientists and physicists like John von Neumann and Alan Turing used his system of algebra for the development of modern computers.

In the latter part of the study you will see the different people who used the Boolean algebra as their channel for the progress of digital computers.

Boole, George (1815-1864), British mathematician and logician, who developed Boolean algebra. He was born on November 2, 1815 in Lincolnshire, England. Largely self-educated, in 1849 Boole was appointed professor of mathematics at Queen’s College (now University College) in Cork, Ireland. In 1854, in An Investigation of the Laws of Thought, Boole described an algebraic system that later became known as Boolean algebra. In Boolean algebra, logical propositions are denoted by symbols and can be acted on by abstract mathematical operators that correspond to the laws of logic. Boolean algebra is of prime importance to the study of pure mathematics and to the design of modern computers. He died on December 8, 1864 in Ballintemple, Country Cork, Ireland.

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Boolean Algebra is branch of mathematics having laws and properties similar to, but different from, those of ordinary high school algebra. Formally a Boolean algebra is a mathematical system consisting of a set of elements, which may be called B, together with two binary operations, which may be denoted by the symbols ⊕ and ⊗. These operations are defined on the set B and satisfy the following axioms:

1. ⊕ and ⊗ are both commutative operations. That is, for any elements x, y of the set B, it is true that x⊕Y = y⊕x and x⊗y = y⊗x.

2. Each of the operations ⊕ and ⊗ distributes over the other. That is, for any elements x, y, and z of the set B, it is true that x⊕ (y⊗z) = (x⊕y) ⊗ (x⊕z), and x⊗ (y⊕z) = (x⊗y) ⊕ (x⊗z).

3. There exists in the set B a distinct identity element for each of the operations ⊕ and ⊗. These elements are usually denoted by the symbols 0 and 1 such that 0 ≠ 1, and have the property that 0 ⊕x = x and 1 ⊗x = x for any element x in the set B.

4. For each element x in the set B there exists a distinct corresponding element called the complement of x, usually denoted by the symbol x’. With respect to the operations ⊕ and ⊗, the element x’ has the property that x⊕x’ = 1 and x ⊗x’ = 0.

A Boolean algebra may have other sets of axioms, all of which may be shown to be equivalent to those just given. The axioms given here are essentially those first published by the American mathematician Edward Huntington in Postulates for the Algebra of Logic (1904).

The English mathematician George Boole gave the first treatment of the subject in 1854. It is possible to denote the operations ⊕ and ⊗ by any two symbols; +, ∨, and ◡ are sometimes used instead of Å, and ×, ^, ∩, ·, and O instead of ⊗.

As an example of a Boolean algebra, consider any set X and let P (X) stand for the collection of all possible subsets of the set X. P (X) is sometimes called the power set of the set X. P (X), together with ordinary set union (◡) and set intersection (∩), forms a Boolean algebra. In fact, every Boolean algebra may be represented as algebra of sets.

From the symmetry of the axioms with respect to the two operations and their respective identities, one is able to prove the so-called principle of duality. This principle asserts that any algebraic statement deducible from the axioms of Boolean algebra remains true if the operations ⊕ and ⊗and the identities 1 and 0 are interchanged throughout the statement. Of the many theorems that can be deduced from the axioms of a Boolean algebra, De Morgan’s laws, that (x⊕y)’ = x’⊗y’ and that (x⊗y)’ = x’⊕y’, are particularly noteworthy.

The elements that are contained in the set B of a Boolean algebra may be abstract objects, or concrete things such as numbers, propositions, sets, or electrical networks. In Boole’s original development, the elements of a Boolean algebra were a collection of propositions, or simple declarative sentences having the property that they were either

true or false but not both. The operations were essentially conjunction and disjunction, denoted by the symbols ^ and ∨ respectively. If x and y represent two propositions, then the expression x∨y (read x or y) would be true if and only if either x or y or both were true. The statement x ^ y (read x and y) would be true if and only if both x and y were true. In this type of Boolean algebra, the complement of an element or proposition is simply the negation of the statement.

A Boolean algebra of propositions and a Boolean algebra of sets are closely connected. For example, let p be the statement, “The ball is blue,” and let P be the set of all elements for which the statement p is true, that is, the set of all blue balls. P is called the truth set for the proposition p. Indeed, if P and Q are the truth sets for statements p and q, then the truth set for the statement p∨q is clearly P ◡ Q and for p ^ q the truth set is P ∩Q.

Boolean algebra has many practical applications in the physical sciences, in electric-circuit theory and particularly in the field of computers.

As an example of an application of Boolean algebra in electrical-circuit theory, let p and q denote two propositions, that is, declarative sentences that are either true or false but not both. If each of the propositions p and q is associated with a switch that will be closed if the proposition is true, and open if the proposition is false, then the statement

p ^ q may be represented by connecting the switches in series. The current will flow in this circuit if and only if both switches are closed, that is, if both p and q are true. Similarly, a circuit with switches connected in parallel can be used to represent the statement p∨q. In this case the current will flow if either p or q or both are true and the respective switches are closed. More complicated statements give rise to more complex switching circuits.

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The objectives of the study aims to know the life of the logician George Boole. To recognize and appreciate the people who used the system of Boolean algebra and the help given by Boole’s logic to the design of modern computers.

The importance of the study is to know whether the logic of George Boole has a great aid to the design of modern computers. To be thankful and grateful to the people that aided and contributed to the design of today’s computers.

This chapter includes the review of Related Literature about the study entitled The Logic of George Boole and its Application to the Design of Modern Computers.

“How do computers do what they do?” and “How does a computer engineer design a computer?” A computer and similar complex machines is at the heart a system of circuits that perform logical and arithmetic operations. It was determined that the cheapest way to make such a machine was to start with the binary number system which could be effectively handled with Boolean algebra. The computer in front of you does its math and logical thinking using Boolean algebra. The Boolean algebra was used to design the switching circuits of the computer. The materials used to build the switching circuits are not important in this lecture. The logical design used to make the switching circuits work correctly is important in this lecture. By using Boolean algebra, the computer can do logical thinking very fast and very inexpensively thanks to the underlying semiconductor electronic circuitry.

In the future computers chips may be made of some material other than semiconductors, perhaps a biological material that is alive and has been genetically designed to do Boolean algebra. And so computers would be grown in laboratories. This

may lead to truly new life forms. I hope that thought doesn’t scare you, but motivates you to learn computer logic. But for now the computers we have today are still the marvels of technology. And even though you find computers of different flavors and capabilities, they all have a few things in common. One very important common thing among computers is that they process information using the rules of Boolean algebra.

http://museum.doorsofperception.com/doors/doors1/transcripts/heim/heim3.html

This chapter includes the Presentation, Analysis and Interpretation of Data about the study.

NAMEYEARCONTRIBUTION TO THE DESIGN OF MODERN COMPUTERS

George BooleBritish logician and mathematician1847Boolean algebra

Konrad ZuseGerman Engineer1930’sZ1 calculating machine

John AtanasoffAmerican PhysicistLate 1930’sDesign of the first digital computer

Clifford BerryAmerican PhysicistLate 1930’sDesign of the first digital computer

Alan TuringBritish mathematician1930’sRecognized binary logic for the development in digital computers

Claude ShannonAmerican Mathematician1940’sRecognized binary logic for the development in the digital computers

John von NeumannHungarian-born Mathematician1944 to 1945Usage of the binary arithmetic system for storing programs in computer

B. Analysis and Interpretation of Data

19th-century British logician and mathematician George Boole, who in 1847 invented a two-valued system of algebra that represented logical relationships and operations first proposed binary logic. German engineer Konrad Zuse used this system of algebra, called Boolean algebra, in the 1930s for his Z1 calculating machine. American physicist John Atanasoff and his graduate student Clifford Berry also used it in the design of the first digital computer in the late 1930s. During 1944 and 1945 Hungarian-born American mathematician John von Neumann suggested using the binary arithmetic system for storing programs in computers. In the 1930s and 1940s British mathematician Alan Turing and American mathematician Claude Shannon also recognized how binary logic was well suited to the development of digital computers.

This chapter includes the summary, conclusion and the recommendation of the study entitled the logic of George Boole and its application to the design of modern computers.

Today in our modern world, people use computers in making their researches and in storing important facts and information. It was made possible by the logic of George Boole, Boolean algebra and the mathematicians like John von Neumann and Claude Shannon. In the year 1847, British logician and mathematician George Boole invented a two-valued system of algebra called Boolean algebra. Then Konrad Zuse who independently developed “electromechanical” computers, in which a series of electrically controlled devices known as relay represented numbers, Z1 calculating machine used Boole’s concept. In late 1930’s, John Atanasoff, Clifford Berry and Alan Turing designed the first digital computer and introduced the concept of a theoretical computing device now known as Turing device, which was important in the development of the digital computer respectively. Claude Shannon who developed information theory, which is a theoretical study, however, it has had a profound impact on the design of practical data communication and storage systems, such as telephones and computers. During 1944 to 1945, Hungarian-born American mathematician, John von Neumann: who was also known for the design of high-speed electronic computers.

These people became a part and have a great part in the advancement in the design of digital computers. This only means that Boole affects all of our lives in this contemporary world together with the people who contributed in the design of modern computers

Therefore I conclude that the concept of Boolean algebra by George Boole really helped in the development and progress in the design of digital computers. Through the people like Alan Turing and Konrad Zuse who aided and used Geroge Booles’ two-valued system of algebra in the advancement of computers that without them digital computers would be impossible. This connection of Boole with the design of computers means only that he affects all of our lives in the modern world because we are using the product of his logic in our every day lives.

I would like to recommend that further studies should be conducted about Boolean algebra and how it works and used as a mathematical system primarily used in the design of computers. The life of George Boole and the people who used his logic should also be deeply studied.

Bibliography:

http://www.encyclopedia.com/html/b/booleanal.asp

http://www.brightplanet.com/deepcontent/tutorials/search/part4.asp

http://museum.doorsofperception.com/doors/doors1/transcripts/heim/heim3.html

Microsoft® Encarta® Reference Library 2003. © 1993-2002 Microsoft Corporation.

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