Boolean Algebra
Boolean algebra is a mathematical system used to analyze and simplify digital circuits. It deals with variables that can have only two possible values, typically represented as 0 and 1. The algebraic operations in Boolean algebra include AND, OR, and NOT, which are used to manipulate and simplify logical expressions. It forms the foundation for digital logic design and is essential in computer science and engineering.
8 Key excerpts on "Boolean Algebra"
eBook - ePubDigital Design
Basic Concepts and Principles
- Mohammad A. Karim, Xinghao Chen(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...2 Boolean Algebra 2.1 Introduction Design of digital circuits is typically approached from an understanding of mathematical formalism pertinent to binary systems. This particular formalism is commonly known as Boolean Algebra. Claude Shannon proposed this particular algebra, by extending the works of algebra of logic that was initially devised by George Boole, for analyzing and designing discrete binary systems. Boolean Algebra is a mathematical system that defines three primary logical operations: AND, OR, and NOT, on sets of binary logical variables. Boole had used his algebraic manipulations for describing the logical relationships of natural language. This reference to natural language is very relevant here since we are also interested in translating a word statement of the function of the desired digital system to a mathematical description. Boolean Algebra serves as the basis for moving from a verbal description of the logical function to an unambiguous mathematical description. This unambiguous representation allows us to design logic circuits using a given library of logic components. Boolean Algebra is finite but richer in scope than the ordinary algebra and, accordingly, it leads to simpler optimization schemes. Complex logical functions can be simplified using Boolean Algebraic rules. Correspondingly, the design process leads to logic circuits that are both simplified and inexpensive. The properties of Boolean Algebra need to be understood first before we could learn to design digital systems efficiently. This chapter will acquaint you with Boolean Algebra and provide necessary tools for handling complex logical functions. 2.2 Logic Operations The logic functions introduced here are the allowed operations in Boolean Algebra, which is explored later in Section 2.4. An understanding of these logic operations is vital since they are used in translating a word statement of a logical problem to a digital logic network...
eBook - ePub- Brian Holdsworth, Clive Woods(Authors)
- 2002(Publication Date)
- Newnes(Publisher)
...2 Boolean Algebra 2.1 Introduction In a digital system the electrical signals that are used have two voltage levels which may, for example, be 5 and 0 volts. The electrical devices used in these systems can generally exist indefinitely in one of these two possible voltage states, providing the power supply is maintained. For example, a bipolar transistor that is non-conducting in a 5 volt system will have approximately 5 volts between collector and emitter. However, when the transistor is turned on and is conducting, it can be arranged, with a suitable choice of load, that the voltage between collector and emitter is approximately zero. The two voltage levels employed in a digital circuit can be arbitrarily assigned values of 0 and 1. The two states defined in this way can have logical significance in that they can indicate the presence of a particular condition or, alternatively, its absence. An algebra developed in the nineteenth century by George Boole (1815-1864), an English mathematician, is well suited for representing the situation above. This branch of mathematics, called Boolean Algebra, is a discrete algebra in which the variables can have one of two values, either 0 or 1. Associated with the algebra is a number of theorems which allow the manipulation and simplification of Boolean equations. Shannon, who was the first to develop information theory, became aware that Boolean Algebra was useful in the design of switching networks. Initially, the algebra was used in the design of relay networks. More recently switching circuits were implemented using discrete components but rapid technological advances have seen the introduction of MSI, LSI and VLSI devices and because of the sophisticated and versatile nature of these components there have been significant changes in the design techniques used by engineers...
eBook - ePubBowtie Methodology
A Guide for Practitioners
- Sasho Andonov(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...4 Boolean Algebra 4.1 Introduction Logic has been established as the science of thought, reasoning, and thinking, since the age of Ancient Greece. Aristotle was one of the first to dedicate his work to the logical thinking. In the middle of the nineteenth century, the British mathematician George Boole published a book with long name, but known today only as “The Laws of Thought.” He was the first mathematician who tried to establish a systematic way in dealing with principles of logic and correct reasoning. That is the reason why the mathematical area of dealing with the quantification of logic operations is named (in his honor) Boolean Algebra. The Boolean Algebra is actually a deductive mathematical system known as the algebra of logic and reasoning, and it had a dramatic influence to the computer development in the twentieth century. Even today, all processors use the Boolean Algebra to conduct logic operations and calculate mathematical expressions. Having in mind that Boole quantified the sentences (expressions) by their value (are they “true” or “false”), he used the binary numerical system to calculate their combinations. Later, this was accepted by computer engineers because the “true” could be expressed as 1 (current is flowing or voltage is present) and “false” could be expressed as 0 (current is not flowing or voltage is not present). The contribution of Boole is also important for the connection of probability with logic. It was a normal development of logic, using the probability of previous events to calculate the probability of future events. So, the Boolean Algebra deals with logic and probability, and it uses the binary numerical system. Actually, using 0 and 1 is not so simple. As I have mentioned earlier, the situation “1” (current is flowing) and “0” (current is not flowing) can be counted as switches. Switch in position ON means 1 (current is flowing) and switch in position OFF means 0 (current is not flowing)...
eBook - ePub- J. Crowe, Barrie Hayes-Gill(Authors)
- 1998(Publication Date)
- Butterworth-Heinemann(Publisher)
...1 Fundamentals 1.1 INTRODUCTION This chapter introduces the essential information required for the rest of the book. This includes a description of Boolean Algebra, the mathematical language of digital electronics, and the logic gates used to implement Boolean functions. Also covered are the ‘tools’ of digital electronics such as truth tables, timing diagrams and circuit diagrams. Finally, certain concepts such as duality, positive and negative assertion level logic and universal gates, that will be used in later chapters, are introduced. 1.2 BASIC PRINCIPLES 1.2.1 Boolean Algebra - an introduction The algebra of a number system basically describes how to perform arithmetic using the operators of the system acting upon the system’s variables which can take any of the allowed values within that system. Boolean Algebra describes the arithmetic of a two-state system and is therefore the mathematical language of digital electronics. The variables in Boolean Algebra are represented as symbols (e.g. A, B, C, X, Y etc.) which indicate the state (e.g. voltage in a circuit). In this book this state will be either 0 or 1. 1 Boolean Algebra has only three operators: NOT, AND and OR. The symbols representing these operations, their usage and how they are used verbally are all shown in Table 1.1. Note that whereas the AND 2 and OR operators operate on two or more variables the NOT operator works on a single variable. Table 1.1 Boolean variables and operators Example 1.1 A circuit contains two variables (i.e. signals), X and Y, which must be OR’d together. How would this operation be shown using Boolean Algebra, and how would you describe it verbally? Solution The operation would be spoken as X or Y and written as X + Y. Example 1.2 The output Y of a logic circuit with two inputs, A and B, is given by the Boolean arithmetic expression,...
eBook - ePub- John Stonham(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...It is, however, of great practical importance as it provides a quick and simple method of determining equality between complex logic systems. Boolean Algebra provides a mathematical foundation for binary information processing. It can be used to describe complex operations, prove identities and simplify logic systems. In practice, however, Boolean Algebra is of limited value to the engineer. The success of a minimization or a proof of an identity depends largely on the person’s expertise, and success cannot be guaranteed. If we take two logic systems where one is considerably more complex than the other, it is important to know whether they perform the same process. If they do, then the engineer would always choose the simpler system. Boolean Algebra can be used to establish identities; however, if two systems cannot be proved equivalent by algebra, the result is inconclusive. Either the systems are not equivalent or the engineer’s algebra is inadequate. A better, more reliable method of demonstrating equivalence between two systems is to generate their respective truth tables. If two or more systems have identical truth tables, then they perform the same function. Worked Example 2.9 If F 1 = A (A + B ¯) + BC (A ¯ + B) + B ¯ (A ⊕ C) and F 2 = A + C determine whether F 1 = F 2 by means of truth tables. Solution F 1 is a function of A, B and C. F 2 can be considered a function of A, B and C with its output independent of B. Truth tables are Remember A⊕C is the EX.OR function and equal to A. C ¯ + A ¯. C. If the output cannot be calculated directly from the Boolean equations and the input values, evaluate any necessary intermediate terms first. By comparing the output columns, we can see that F 1 and F 2 always have the same value for each and every output. Hence F 1 = F 2. Exercise 2.2 Use Boolean Algebra to determine whether or not the following functions are...
eBook - ePub- J. Gibson(Author)
- 2013(Publication Date)
- Routledge(Publisher)
...Boolean Algebra is similar but it is based on the arithmetic of logic values. In Boolean arithmetic numeric quantities may only have either of the two logic values true and false; these have already been given alternative names of 1 and 0. Instead of the four conventional arithmetical operations of add, subtract, multiply and divide the three Boolean operations of AND, OR and NOT are used. These operations have already been defined and are summarized in Table 2.10. Equivalence or identity symbols are used to indicate that terms on each side of a relationship arc identical. For example 1 + 0 = 1 indicates that 1 + 0 may be replaced by 1 as the result is identical; however it is also the case that 1 may be replaced by 1 + 0 (1 may also be replaced in many other ways). Each of the relationships in Table 2.10 may be shown to be correct by applying the definition of a particular Boolean operation. Table 2.10 Boolean arithmetic operations The relations in Table 2.10 indicate that the order of evaluating terms in a Boolean operation is not important, as indicated 0 + 1 ≡ 1 + 0 and 0.1 ≡ 1.0. Relationships involving a single Boolean operation may be extended to any number of terms; either directly from the definition of the Boolean operations, or by using identities in Table 2.10. For example using 1 ≡ 1 + 0 to replace the left hand 1 of 1 + 1 ≡ 1 by its equivalent 1 + 0 produces 1 + 0 + 1 ≡ 1. Extension to other cases is trivial and may be used to demonstrate that the order of terms in a Boolean operation has no effect; results such as the following are easily obtained These may also be derived by logical argument based on the fundamental definitions of the operators AND and OR. In more complex Boolean expressions, that is those that involve more than one of the operators AND, OR and NOT, it is necessary to consider the order of evaluation...
eBook - ePub- Phillip A. Laplante, Phillip A. Laplante(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...Boolean Algebras A. R. Bednarek University of Florida, Gainesville, Florida, U.S.A. Abstract Boolean Algebra, named after the 19 th century mathematician and logician, George Boole, has contributed to many aspects of computer science and information science. In information science, Boolean logic forms the basis of most end-user search systems, from searches in online databases and catalogs, to uses of search engines in information seeking on the World Wide Web. — ELIS Classic, from 1970 Keywords: Boolean Algebra Boole; George. INTRODUCTION In this entry attention is focused on mathematical models of proven utility in the area of information handling, namely, Boolean Algebras. Following some general comments concerning mathematical models, particular examples of Boolean Algebras, serving as motivation for the subsequent axiomatization, are presented. Some elementary theorems are cited, particularly the very important representation theorem that justifies, in some sense, the focusing of attention on a particular Boolean Algebra, namely, the algebra of classes, and applications more directly related to the information sciences are given. Running the risk of redundancy, attention will be called to an often-repeated observation, but one of extreme importance in applications of mathematics to physical problems. Referring to Fig. 1, it is important to realize that when one constructs a mathematical model as a representation of a physical phenomenon, one is abstracting and, as a consequence, the model formulated is doomed to imperfection. That is, one can never formally mirror the physical phenomenon, and must always be satisfied with an imperfect copy. However, following the initial commitment to a model, the logic that one appeals to dictates the resultant theorems derived within the framework of the model...
eBook - ePub- K.C. Hignett(Author)
- 2002(Publication Date)
- Routledge(Publisher)
...Boolean Algebra is deterministic in that it postulates binary states such that a condition is either present or not present, and hence relates to discrete probabilities of 1 and 0. In the practical engineering world it is not possible to have knowledge of a definite elemental state but rather its probability of being ‘present’ or ‘not present’, which implies a probabilistic hypothesis of being somewhere between 1 and 0. To construct a truth table it is necessary to list in appropriate columns the defined elemental states such that taken together they express all the possible resultant system states which they produce. Truth tables may be drawn up for any number of elements but become somewhat large above six. Table 6.1 shows a truth table for three elements A, B and C. When considering Boolean relationships the respective element states are shown as either: Present = A or Not present = A Table 6.1 Truth table - three elements in present/not present logic A B C A B C A B C A B C A B C A B C A B C A B C A B C It is also important to define whether the ‘present’ state relates to failure or success. In Table 6.1 ‘present’ refers to success, hence ‘not present’ refers to the presence of failure. Evidence of correct compilation is verified whereby the table commences with the top row showing all three element states as ‘present’ whilst the bottom row shows all the three states as ‘not present’. The nomenclature of Table 6.1 is most convenient for the Boolean deterministic approach which is pertinent to fault trees. Mathematical modelling, which is dealt with in Chapter 8, also relies on the same truth table in order to derive probabilistic expressions relating to a particular model. For convenience, Table 6.2 suggests a recommended modification of the Boolean-based table (Table 6.1) in order to promote conformity with the probabilistic stage requirements...
