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Boolean Algebra Topics
This is a list of topics around Boolean algebra and propositional logic. Articles with a wide scope and introductions * Algebra of sets * Boolean algebra (structure) * Boolean algebra * Field of sets * Logical connective * Propositional calculus Boolean functions and connectives * Ampheck * Analysis of Boolean functions * Balanced Boolean function * Bent function * Boolean algebras canonically defined * Boolean function * Boolean matrix * Boolean-valued function * Conditioned disjunction * Evasive Boolean function * Exclusive or * Functional completeness * Logical biconditional * Logical conjunction * Logical disjunction * Logical equality * Logical implication * Logical negation * Logical NOR * Lupanov representation * Majority function * Material conditional * Minimal axioms for Boolean algebra * Peirce arrow * Read-once function * Sheffer stroke * Sole sufficient operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Of Sets
In mathematics, the algebra of sets, not to be confused with the mathematical structure of ''an'' algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being ''union'', the meet operator being ''intersection'', the complement operator being ''set complement'', the bottom being and the top being the universe set under consideration. Fundamentals The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditioned Disjunction
In logic, conditioned disjunction (sometimes called conditional disjunction) is a ternary logical connective introduced by Church. Given operands ''p'', ''q'', and ''r'', which represent truth-valued propositions, the meaning of the conditioned disjunction is given by : , q, r\Leftrightarrow (q \to p) \land (\neg q \to r). In words, is equivalent to: "if ''q'', then ''p'', else ''r''", or "''p'' or ''r'', according as ''q'' or not ''q''". This may also be stated as "''q'' implies ''p'', and not ''q'' implies ''r''". So, for any values of ''p'', ''q'', and ''r'', the value of is the value of ''p'' when ''q'' is true, and is the value of ''r'' otherwise. The conditioned disjunction is also equivalent to : (q \land p) \lor (\neg q \land r) and has the same truth table as the ternary conditional operator ?: in many programming languages (with , a, c/math> being equivalent to a ? b : c). In electronic logic terms, it may also be viewed as a single-bit multiplexer. In conjunction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Material Conditional
The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol \to is interpreted as material implication, a formula P \to Q is true unless P is true and Q is false. Material implication is used in all the basic systems of classical logic as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages. However, many logics replace material implication with other operators such as the strict conditional and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentences in natural language. Notation In logic and related fields, the material conditional is customarily notated with an infix operator \to. The material conditional is also notated using the i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Majority Function
In Boolean logic, the majority function (also called the median operator) is the Boolean function that evaluates to false when half or more arguments are false and true otherwise, i.e. the value of the function equals the value of the majority of the inputs. Boolean circuits A ''majority gate'' is a logical gate used in circuit complexity and other applications of Boolean circuits. A majority gate returns true if and only if more than 50% of its inputs are true. For instance, in a full adder, the carry output is found by applying a majority function to the three inputs, although frequently this part of the adder is broken down into several simpler logical gates. Many systems have triple modular redundancy; they use the majority function for majority logic decoding to implement error correction. A major result in circuit complexity asserts that the majority function cannot be computed by AC0 circuits of subexponential size. Properties For any ''x'', ''y'', and ''z'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lupanov Representation
Lupanov's (''k'', ''s'')-representation, named after Oleg Lupanov, is a means of representing Boolean circuits to demonstrate an asymptotically tight upper bound on the circuit size (i.e., the number of gates) needed to represent a Boolean function. Claude Shannon showed that almost all Boolean functions of ''n'' variables need a circuit of size at least 2''n''''n''−1. Lupanov's (''k'', ''s'')-representation shows that all Boolean functions of ''n'' variables can be computed with a circuit of 2''n''''n''−1 + o(2''n''''n''−1) gates. Definition The idea is to represent the values of a boolean function ''ƒ'' in a table of 2''k'' rows, representing the possible values of the ''k'' first variables ''x''1, ..., ,''x''''k'', and 2''n''−''k'' columns representing the values of the other variables. Let ''A''1, ..., ''A''''p'' be a partition of the rows of this table such that for ''i'' < ''p'', , ''A'' [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical NOR
In Boolean logic, logical NOR, non-disjunction, or joint denial is a truth-functional operator which produces a result that is the negation of logical or. That is, a sentence of the form (''p'' NOR ''q'') is true precisely when neither ''p'' nor ''q'' is true—i.e. when both ''p'' and ''q'' are ''false''. It is logically equivalent to \neg(p \lor q) and \neg p \land \neg q, where the symbol \neg signifies logical negation, \lor signifies OR, and \land signifies AND. Non-disjunction is usually denoted as \downarrow or \overline or X (prefix) or \operatorname. As with its dual, the NAND operator (also known as the Sheffer stroke—symbolized as either \uparrow, \mid or /), NOR can be used by itself, without any other logical operator, to constitute a logical formal system (making NOR functionally complete). The computer used in the spacecraft that first carried humans to the moon, the Apollo Guidance Computer, was constructed entirely using NOR gates with three inputs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Negation
In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \overline. It is interpreted intuitively as being true when P is false, and false when P is true. For example, if P is "Spot runs", then "not P" is "Spot does not run". An operand of a negation is called a ''negand'' or ''negatum''. Negation is a unary logical connective. It may furthermore be applied not only to propositions, but also to notions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes ''truth'' to ''falsity'' (and vice versa). In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P is the proposition whose proofs are the refutations of P. Definition ''Classical negation'' is an operation on one logical value, typically th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Implication
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Equality
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work." Premise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Disjunction
In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is sunny or it is warm" can be represented in logic using the disjunctive formula S \lor W , assuming that S abbreviates "it is sunny" and W abbreviates "it is warm". In classical logic, disjunction is given a truth functional semantics according to which a formula \phi \lor \psi is true unless both \phi and \psi are false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is an ''inclusive'' interpretation of disjunction, in contrast with exclusive disjunction. Classical proof theoretical treatments are often given in terms of rules such as disjunction introduction and disjunction elimination. Disjunction has also been given numerous non-classical treatments, motivated by problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Conjunction
In logic, mathematics and linguistics, ''and'' (\wedge) is the Truth function, truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as \wedge or \& or K (prefix) or \times or \cdot in which \wedge is the most modern and widely used. The ''and'' of a set of operands is true if and only if ''all'' of its operands are true, i.e., A \land B is true if and only if A is true and B is true. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In natural language, the denotation of expressions such as English language, English "Conjunction (grammar), and"; * In programming languages, the Short-circuit evaluation, short-circuit and Control flow, control structure; * In set theory, Intersection (set theory), intersection. * In Lattice (order), lattice theory, logical conjunction (Infimum and supremum, greatest lower bound). Notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Biconditional
In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or bidirectional implication or biimplication or bientailment, is the logical connective used to conjoin two statements P and Q to form the statement "P if and only if Q" (often abbreviated as "P iff Q"), where P is known as the ''antecedent (logic), antecedent'', and Q the ''consequent''. Nowadays, notations to represent equivalence include \leftrightarrow,\Leftrightarrow,\equiv. P\leftrightarrow Q is logically equivalent to both (P \rightarrow Q) \land (Q \rightarrow P) and (P \land Q) \lor (\neg P \land \neg Q) , and the XNOR gate, XNOR (exclusive NOR) Logical connective, Boolean operator, which means "both or neither". Semantically, the only case where a logical biconditional is different from a material conditional is the case where the hypothesis (antecedent) is false but the conclusion (consequent) is true. In this case, the result is true for the conditional, but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
