Boolean Logic
Boolean logic is a fundamental concept in computer science that deals with the manipulation of true and false values. It is based on the principles of Boolean algebra, which uses logical operators such as AND, OR, and NOT to make decisions and perform calculations. Boolean logic forms the basis for programming and designing digital circuits.
8 Key excerpts on "Boolean Logic"
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 - 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 - ePubSemiconductor Basics
A Qualitative, Non-mathematical Explanation of How Semiconductors Work and How They are Used
- George Domingo(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...11 Logic Circuits OBJECTIVES OF THIS CHAPTER Now we are going back to talk about how we use semiconductor devices to perform useful operations. In the first two sections we'll talk about the way we interphase with the computer and the basic language we use, that is, Boolean algebra and logic symbols. Then I will explain how we implement this algebra with switches (to clear the concepts) and semiconductor devices, and we'll see how to do arithmetic operations, sums, subtractions, multiplications, and divisions with the devices we covered in the previous chapters. In the previous chapters I very much concentrated on the performance of individual components and how to use them, mainly in the analogue mode. Now we are going digital. 11.1 Boolean Algebra Everybody knows that digital computers work with 1s and 0s. Computers do not understand the number 3. All the computations that computers have to carry out to give us any meaningful results are done using the ON and OFF conditions, ON for 1 and OFF for 0. The large TV screen that give us beautiful and sharp pictures with bright colors is based on millions of points that can be ON or OFF. Each point of light consists of three miniscule LEDs of three different colors. How a computer manipulates all this data is based on the concepts of Boolean algebra. (Forget about the word algebra. There are no equations involved outside of adding and subtracting.) Mr. George Boole (1815–1864; Figure 11.1) was a British mathematician and philosopher with interest in strengthening the logic concepts. Aristotle is credited with creating logic thinking with his famous syllogisms, such as “All men are mortal, I am a man, therefore I am mortal” or, more abstractly, “All A are B, all B are C, therefore all A are C”. He wanted to be sure that people were logical and consistent in advancing any idea. What Boole did was to add mathematical formalism, symbolic logic, to Aristotle's logic...
eBook - ePub- David Johnson(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...2 Logic Concepts Whenever the nature of the subject permits the reasoning process to be without danger carried on mechanically, the language should be constructed on as mechanical principles as possible… George Boole 2.1 BINARY LOGIC–THE ON/OFF CONCEPT In control systems found using programmable controllers, the preponderance of sensors and actuators are concerned with just two states: ON or OFF. In logic circuits, this corresponds to true/false, or in symbols, A and A’. To relate this to control sytems, an input to the programmable control system from a pushbutton has but two states: ON or OFF. That is, either the button is being pushed, or it isn’t. Once the state or change of state is registered in the programmable controller memory, it can be used in any number of ways, but the principle is fundamental. This is the basic principle of binary logic, binary meaning consisting of two things or parts. The binary numbering system is based on the number two, and hence all sets are represented by combinations of 0 and 1. For example, while it is clear that 00 in binary is 0, and 01 is 1, it may not be clear that 10 equals 2, and 11 equals 3. We will examine these extensions of the binary numbering systems in details in Chapter 3. For now we will concern ourselves with the convention only that the 0 and 1 of binary logic correspond to the OFF and ON of control circuit inputs and outputs. To see this relationship more clearly, consider Figure 2.1(a). Here we have a simple control circuit consisting of two components: a switch A (perhaps from a pushbutton) and a load M. Assume that a sufficient voltage exists across the two vertical lines to power the load M. It should be clear that when the contacts on switch A change state and close, current will pass through it to the load M, thereby energizing that load. This is the fundamental binary variable case and illustrates the identity function A = M. Figure 2.1(b) shows the Venn Diagram for this same identity...
eBook - ePubElectronics
from Classical to Quantum
- Michael Olorunfunmi Kolawole(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...2 Functional Logics Many tasks in modern computer, communications, and control systems are performed by logic circuits. Logic circuits are made of gates. A logic gate is a physical device that implements a Boolean function that performs a logic operation on one or more logical inputs (or terminals) and produces a single logical output. This chapter examines the basic principles of logic gates: the types—from primitive to composite gates—and how they are arranged to perform basic and complex functions. 2.1 The Logic of a Switch Basic logic circuits with one or more inputs and one output are known as gates. Logic gates (or simply gates) are used as the building blocks in the design of more complex digital logic circuits. A logic gate is a physical device that implements a Boolean function that performs a logic operation on one or more logical inputs (or terminals) and produces a single logical output. Practically, gates function by “opening” or “closing” to allow or reject the flow of digital information. For any given moment, every terminal is in one of the two binary conditions “0” (low) or “1” (high). These binary conditions represent different voltage levels; that is, any voltage v up to the device threshold voltage, V th, (i.e. 0 ≤ v ≤ V th); and in the conduction ranges 0 ≤ v ≤ 2.5 V and 2.5 < v ≤ 5 V represent logic states “0” and “1,” respectively. Note that machine arithmetic is accomplished in a two-value (binary) number system, but Boolean algebra—a two-value symbolic logic system—is not a generalization of the binary number system. The symbols 0, 1, +, and • are used in both systems but with totally different meanings in each system. (The meanings of these symbols become obvious during discussions in the next paragraphs.) This symbolic tool allows us to design complex logic systems with the certainty that they will carry out their function exactly as intended...
eBook - ePub- John Stonham(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...A simple sequential circuit has memory properties and finally a memory circuit can be organized as a combinational logic function. These applications will be examined in the following chapters. The fundamental electronic circuits common to all types of logic systems are the gates. A logic function is implemented with one or more gates and the relationships between functions are governed by the laws of Boolean algebra. Logic and propositional statements The formal analysis of binary systems was first investigated by the nineteenth century English mathematician, George Boole, long before the advent of electronics and computers. Boole’s logical algebra was developed to test the validity of propositions and verbal statements, but his theorems are directly relevant to the operations which are performed on binary data in electronic systems. Consider the simple statement about the weather: ‘It will snow if the temperature is low and it is cloudy.’ The three variables are ‘snow’ (the output), ‘low temperature’ and ‘cloudy’ (the inputs). Each of these conditions can only be true or false (i.e. true if it is snowing or false if it isn’t). This weather system can be described by a Boolean equation, with the variables S for snow, L for low temperature and C for cloudy, giving S = L AND C (2.3) The logic function between the two input variables is AND, which can be identified in the verbal statement. The symbol for AND is. and equation 2.3 would normally be written S = L. C (2.4) The symbol. may be omitted. S = LC is the same as S = L.C. The equation states that S is true if, and only if, L is true AND C is true. It represents the proposition ‘It will snow if the temperature is low and it is cloudy.’ Truth tables Every Boolean equation has a truth table that lists the value of the output for each and every possible combination of inputs. If the output is a function of two variables there are 2 2 or four possible input combinations...
eBook - ePub- Raymond F. Gardner(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...This chapter is limited to digital-input and digital-output signal processing. With digital control, there are two types of logic, combinational and sequential. Combinational logic depends on the present state of input signals, where the input originates from switches that are either open or closed, and which in turn sets the output to be either on or off. In contrast, sequential logic uses the same digital-input signals combined with the output signals produced from the previous state to create an updated output signal. Often, sequential logic uses edge-triggered pulses to enable the sequence. The pulses may be synchronously generated in a repetitive manner from an electronic oscillator, called a “clock,” or the pulses may be asynchronously initiated by external switches at random times, such as from manual actuation. Sequential logic needs a built-in form of memory to produce an output that depends on the prior state. There are several ways to describe logic strategies, including logic gates, flow diagrams having if-then conditional statements, truth tables, and relay logic. Logic schemes can be complicated, but often they are broken into simpler, more manageable modules, procedures, or blocks of code. Boolean Logic Boolean Logic is named after George Boole, and uses two-state inputs and two-state outputs in a logic-gate decision-making process. The two states are often described as “true” or “false,” and are commonly represented as “1” or “0.” Conditional states are passed into logic gates where they are manipulated into a single true/false or 1/0 output. In Boolean Logic, there are seven types of simple logic gates. The three primary gates are AND, OR, and NOT gates and are shown in Figure 4.1. The three primary gates form building blocks for creating the remaining four gates. The NOT gate is simply an inverter that toggles a single input to its opposite value as its output...
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...
