|
Algebraic Normal Form
In Boolean algebra, the algebraic normal form (ANF), ring sum normal form (RSNF or RNF), '' Zhegalkin normal form'', or ''Reed–Muller expansion'' is a way of writing logical formulas in one of three subforms: * The entire formula is purely true or false: *: 1 *: 0 * One or more variables are combined into a term by AND (\and), then one or more terms are combined by XOR (\oplus) together into ANF. Negations are not permitted: : a \oplus b \oplus \left(a \and b\right) \oplus \left(a \and b \and c\right) * The previous subform with a purely true term: : 1 \oplus a \oplus b \oplus \left(a \and b\right) \oplus \left(a \and b \and c\right) Formulas written in ANF are also known as Zhegalkin polynomials and Positive Polarity (or Parity) Reed–Muller expressions (PPRM). Common uses ANF is a normal form, which means that two equivalent formulas will convert to the same ANF, easily showing whether two formulas are equivalent for automated theorem proving. Unlike other normal f ... [...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 variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (''and'') denoted as ∧, disjunction (''or'') denoted as ∨, and the negation (''not'') denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction and division. So Boolean algebra is 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 Huntington, the term "Boolean alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Property
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, one has 2 \cdot (1 + 3) = (2 \cdot 1) + (2 \cdot 3). One says that multiplication ''distributes'' over addition. This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted \,\land\,) and the logical or (denoted \,\lor\,) distributes over the other. Definition Given a set S and two binary operators \,*\, and \,+\, on S, *the operation \,*\, is over (or with respect to) \,+\, if, given any elements x, y, \text z of S, x * (y + z) = (x * y) + (x * z); *the operation \,*\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
EETimes
''EE Times'' (''Electronic Engineering Times'') is an electronics industry magazine published in the United States since 1972. EE Times is currently owned by AspenCore, a division of Arrow Electronics since August 2016. Since its acquisition by AspenCore, EE Times has seen major editorial and publishing technology investment and a renewed emphasis on investigative coverage. New features include The Dispatch, which profiles frontline engineers and unpacks the real-life design problems and their solutions in technical yet conversational reporting. Ownership and status ''EE Times'' was launched in 1972 by Gerard G. Leeds of CMP Publications Inc. In 1999, the Leeds family sold CMP to United Business Media for $900 million. After 2000, ''EE Times'' moved more into web publishing. The shift in advertising from print to online began to accelerate in 2007 and the periodical shed staff to adjust to the downturn in revenue. In July 2013, the digital edition migrated to UBM TechWeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Duisburg-Essen
The University of Duisburg-Essen (german: link=no, Universität Duisburg-Essen) is a public research university in North Rhine-Westphalia, Germany. In the 2019 ''Times Higher Education World University Rankings'', the university was awarded 194th place in the world. It was originally founded in 1654 and re-established on 1 January 2003, as a merger of the Gerhard Mercator University of Duisburg and the university of Essen. It is based in both the cities of Duisburg and Essen, and a part of University Alliance Metropolis Ruhr. With its 12 departments and around 40,000 students, the University of Duisburg-Essen is among the 10 largest German universities. Since 2014, research income has risen by 150 percent. Natural science and engineering are ranked within the top 10 in Germany, and the humanities are within the top 20 to 30. Especially, the physics field is ranked in the top 1 in Germany. History Origins: University of Duisburg (1555) The university's origins date ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiley-Teubner
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, technical, and engineering subject areas, abandoning its literary interests. Wiley's son John (born in Flatbush, New York, October 4, 1808; died in East Orange, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science + Business Media B
Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinational academic publishing group created by the merger of Springer Science+Business Media, Nature Publishing Group, Palgrave Macmillan, and Macmillan Education * Axel Springer SE, an important conservative German publishing house, including several newspapers * Springer Publishing Company, an American publishing company of academic journals and books, focusing on public health and the like Places ;United States * Springer, New Mexico * Springer, Oklahoma * Springer Mountain, southern terminus of the Appalachian Trail * Springer Opera House, Columbus, Georgia Animals * In cattle, a cow or heifer near to calving * English Springer Spaniel, a breed of dog * Welsh Springer Spaniel, a breed of dog * Springer (orca), a wild orca (killer whale) also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Ring
In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of 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 logical disjunction, 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. Notations 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 ''x'' + ''y'' = (''x'' ∧ ¬ ''y'') ∨ (¬ ''x'' ∧ ''y'') for the ring sum of ''x'' and ''y'', and use ''xy'' = ''x'' ∧ ''y'' for their product. *In M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karnaugh Map
The Karnaugh map (KM or K-map) is a method of simplifying Boolean algebra expressions. Maurice Karnaugh introduced it in 1953 as a refinement of Edward W. Veitch's 1952 Veitch chart, which was a rediscovery of Allan Marquand's 1881 ''logical diagram'' aka Marquand diagram but with a focus now set on its utility for switching circuits. Veitch charts are also known as Marquand–Veitch diagrams or, rarely, as Svoboda charts, and Karnaugh maps as Karnaugh–Veitch maps (KV maps). The Karnaugh map reduces the need for extensive calculations by taking advantage of humans' pattern-recognition capability. It also permits the rapid identification and elimination of potential race conditions. The required Boolean results are transferred from a truth table onto a two-dimensional grid where, in Karnaugh maps, the cells are ordered in Gray code, and each cell position represents one combination of input conditions. Cells are also known as minterms, while each cell value represents th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunctive Normal Form
In boolean logic, a disjunctive normal form (DNF) is a canonical normal form of a logical formula consisting of a disjunction of conjunctions; it can also be described as an OR of ANDs, a sum of products, or (in philosophical logic) a ''cluster concept''. As a normal form, it is useful in automated theorem proving. Definition A logical formula is considered to be in DNF if it is a disjunction of one or more conjunctions of one or more literals. A DNF formula is in full disjunctive normal form if each of its variables appears exactly once in every conjunction. As in conjunctive normal form (CNF), the only propositional operators in DNF are and (\wedge), or (\vee), and not (\neg). The ''not'' operator can only be used as part of a literal, which means that it can only precede a propositional variable. The following is a context-free grammar for DNF: # ''DNF'' → (''Conjunction'') \vee ''DNF'' # ''DNF'' → (''Conjunction'') # ''Conjunction'' → ''Literal'' \wedge ''Conju ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjunctive Normal Form
In Boolean logic, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs. As a canonical normal form, it is useful in automated theorem proving and circuit theory. All conjunctions of literals and all disjunctions of literals are in CNF, as they can be seen as conjunctions of one-literal clauses and conjunctions of a single clause, respectively. As in the disjunctive normal form (DNF), the only propositional connectives a formula in CNF can contain are and, or, and not. The not operator can only be used as part of a literal, which means that it can only precede a propositional variable or a predicate symbol. In automated theorem proving, the notion "''clausal normal form''" is often used in a narrower sense, meaning a particular representation of a CNF formula as a set of sets of literals. Examples and non-exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Graph
A logical graph is a special type of diagrammatic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. In his papers on ''qualitative logic'', ''entitative graphs'', and '' existential graphs'', Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic. In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures. See also * Charles Sanders Peirce bibliography * Conceptual graph * Propositional calculus * Truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ... * Venn diagram External links {{commonscat-inline, Logica ... [...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 formul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
