|
Boolean Operations In Computer-aided Design
Any kind of logic, function, expression, or theory based on the work of George Boole is considered Boolean. Related to this, "Boolean" may refer to: * Boolean data type, a form of data with only two possible values (usually "true" and "false") * Boolean algebra, a logical calculus of truth values or set membership * Boolean algebra (structure), a set with operations resembling logical ones * Boolean domain, a set consisting of exactly two elements whose interpretations include ''false'' and ''true'' * Boolean circuit, a mathematical model for digital logical circuits. * Boolean expression, an expression in a programming language that produces a Boolean value when evaluated * Boolean function, a function that determines Boolean values or operators * Boolean model (probability theory), a model in stochastic geometry * Boolean network, a certain network consisting of a set of Boolean variables whose state is determined by other variables in the network * Boolean processor, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Boole
George Boole ( ; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. He worked in the fields of differential equations and algebraic logic, and is best known as the author of ''The Laws of Thought'' (1854), which contains Boolean algebra. Boolean logic, essential to computer programming, is credited with helping to lay the foundations for the Information Age. Boole was the son of a shoemaker. He received a primary school education and learned Latin and modern languages through various means. At 16, he began teaching to support his family. He established his own school at 19 and later ran a boarding school in Lincoln. Boole was an active member of local societies and collaborated with fellow mathematicians. In 1849, he was appointed the first professor of mathematics at Queen's College, Cork (now University C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Data Type
In computer science, the Boolean (sometimes shortened to Bool) is a data type that has one of two possible values (usually denoted ''true'' and ''false'') which is intended to represent the two truth values of logic and Boolean algebra. It is named after George Boole, who first defined an algebraic system of logic in the mid 19th century. The Boolean data type is primarily associated with conditional statements, which allow different actions by changing control flow depending on whether a programmer-specified Boolean ''condition'' evaluates to true or false. It is a special case of a more general ''logical data type—''logic does not always need to be Boolean (see probabilistic logic). Generalities In programming languages with a built-in Boolean data type, such as Pascal, C, Python or Java, the comparison operators such as > and ≠ are usually defined to return a Boolean value. Conditional and iterative commands may be defined to test Boolean-valued expressions. L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denoted by 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as Logical conjunction, conjunction (''and'') denoted as , disjunction (''or'') denoted as , and negation (''not'') denoted as . Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal way of describing logical operations in the same way that elementary algebra describes numerical operations. Boolean algebra was introduced by George Boole in his first book ''The Mathematical Analysis of Logic'' (1847), and set forth more fully in his ''An Investigation of the Laws of Thought'' (1854). According to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle. __TOC__ History The term "Boolean algebra" honors George Boole (1815–1864), a self-educated E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Domain
In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as , or \mathbb. The algebraic structure that naturally builds on a Boolean domain is the Boolean algebra with two elements. The initial object in the category of bounded lattices is a Boolean domain. In computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programming languages feature reserved words or symbols for the elements of the Boolean domain, for example false and true. However, many programming languages do not have a Boolean data type in the strict sense. In C or BASIC, for example, falsity is represented by the number 0 and truth is represented by the number 1 or −1, and all variables that can take these values can also take any other numerical values. Generalizations Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Circuit
In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal language can be decided by a family of Boolean circuits, one circuit for each possible input length. Boolean circuits are defined in terms of the logic gates they contain. For example, a circuit might contain binary AND and OR gates and unary NOT gates, or be entirely described by binary NAND gates. Each gate corresponds to some Boolean function that takes a fixed number of bits as input and outputs a single bit. Boolean circuits provide a model for many digital components used in computer engineering, including multiplexers, adders, and arithmetic logic units, but they exclude sequential logic. They are an abstraction that omits many aspects relevant to designing real digital logic circuits, such as metastability, fanout, glitches, power consumption, and propagation delay variability. Formal definition In givi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Expression
In computer science, a Boolean expression (also known as logical expression) is an expression used in programming languages that produces a Boolean value when evaluated. A Boolean value is either true or false. A Boolean expression may be composed of a combination of the Boolean constants True/False or Yes/No, Boolean-typed variables, Boolean-valued operators, and Boolean-valued functions. Boolean expressions correspond to propositional formulas in logic and are associated to Boolean circuits. Boolean operators Most programming languages have the Boolean operators OR, AND and NOT; in C and some languages inspired by it, these are represented by ", , " (double pipe character), "&&" (double ampersand) and "!" ( exclamation point) respectively, while the corresponding bitwise operations are represented by ", ", "&" and "~" (tilde).E.g. for Java see . In the mathematical literature the symbols used are often "+" ( plus), "·" ( dot) and overbar, or "∨" ( vel), "∧" ( et ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Function
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean functions are the subject of Boolean algebra and switching theory. A Boolean function takes the form f:\^k \to \, where \ is known as the Boolean domain and k is a non-negative integer called the arity of the function. In the case where k=0, the function is a constant element of \. A Boolean function with multiple outputs, f:\^k \to \^m with m>1 is a vectorial or ''vector-valued'' Boolean function (an S-box in symmetric cryptography). There are 2^ different Boolean functions with k arguments; equal to the number of different truth tables with 2^k entries. Every k-ary Boolean function can be expressed as a propositional formula in k variables x_1,...,x_k, and two propositional formulas a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Model (probability Theory)
For statistics in probability theory, the Boolean-Poisson model or simply Boolean model for a random subset of the plane (or higher dimensions, analogously) is one of the simplest and most tractable models in stochastic geometry. Take a Poisson point process of rate \lambda in the plane and make each point be the center of a random set; the resulting union of overlapping sets is a realization of the Boolean model . More precisely, the parameters are \lambda and a probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ... on compact sets; for each point \xi of the Poisson point process we pick a set C_\xi from the distribution, and then define as the union \cup_\xi (\xi + C_\xi) of translated sets. To illustrate tractability with one simple formula, the mean de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Network
A Boolean network consists of a discrete set of Boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done synchronously or asynchronously. Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly convey the correct pattern of expressed and suppressed genes. The se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Processor
The Intel MCS-51 (commonly termed 8051) is a single-chip microcontroller (MCU) series developed by Intel in 1980 for use in embedded systems. The architect of the Intel MCS-51 instruction set was John H. Wharton.. Intel's original versions were popular in the 1980s and early 1990s, and enhanced binary compatible derivatives remain popular today. It is a complex instruction set computer with separate memory spaces for program instructions and data. Intel's original MCS-51 family was developed using N-type metal–oxide–semiconductor ( NMOS) technology, like its predecessor Intel MCS-48, but later versions, identified by a letter C in their name (e.g., 80C51) use complementary metal–oxide–semiconductor ( CMOS) technology and consume less power than their NMOS predecessors. This made them more suitable for battery-powered devices. The family was continued in 1996 with the enhanced 8-bit MCS-151 and the 8/ 16/32-bit MCS-251 family of binary compatible microcontrollers. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Ring
In mathematics, a Boolean ring is a ring for which for all in , that is, a ring that consists of only idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet , and ring addition to exclusive disjunction or symmetric difference (not disjunction , which would constitute a semiring). Conversely, every Boolean algebra gives rise to a Boolean ring. Boolean rings are named after the founder of Boolean algebra, George Boole. Notation There are at least four different and incompatible systems of notation for Boolean rings and algebras: * In commutative algebra the standard notation is to use for the ring sum of and , and use for their product. * In logic, a common notation is to use for the meet (same as the ring product) and use for the join, given in terms of ring notation (given just above) by . * In set theory and logic it is also common to use f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
