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Encyclopedia > Logical disjunction

In logic and mathematics, logical disjunction (written or) is a logical operator that results in true just whenever one or more of its operands are true. If the values 0 and 1 are used for false and true respectively, then: Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... A logic gate performs a logical operation on one or more logic inputs and produces a single logic output. ... Image File history File links Logical_disjunction. ... Image File history File links Sound-icon. ... For the Manfred Mann album, see 2006 (album). ... July 21 is the 202nd day (203rd in leap years) of the year in the Gregorian calendar, with 163 days remaining. ... Image File history File links Sound-icon. ... Logic, from Classical Greek λόγος logos (meaning word, account, reason or principle), is the study of the principles and criteria of valid inference and demonstration. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ...

a vee b = a + b - ab.

Contents

Definition

Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false. In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ... In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ...


The truth table of p OR q (also written as p ∨ q) is as follows: Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

Logical Disjunction
p q p ∨ q
F F F
F T T
T F T
T T T

More generally a disjunction is a logical formula that can have one or more literals separated only by ORs. A single literal is often considered to be a degenerate disjunction. Look up literal, literally in Wiktionary, the free dictionary. ...


Symbol

The mathematical symbol for logical disjunction varies in the literature. In addition to the word "or", the symbol "∨", deriving from the Latin word vel for "or", is commonly used for disjunction. For example: "AB " is read as "A or B ". Such a disjunction is false if both A and B are false. In all other cases it is true.


All of the following are disjunctions:

AB
¬AB
A ∨ ¬B ∨ ¬CD ∨ ¬E

The corresponding operation in set theory is the set-theoretic union. In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...


Associativity and commutativity

For more than two inputs, or can be applied to the first two inputs, and then the result can be or'ed with each subsequent input:

(A or (B or C)) ⇔ ((A or B) or C)

Because or is associative, the order of the inputs does not matter: the same result will be obtained regardless of association. In mathematics, associativity is a property that a binary operation can have. ...


The operator or is also commutative and therefore the order of the operands is not important: A map or binary operation from a set to a set is said to be commutative if, (A common example in school-math is the + function: , thus the + function is commutative) Otherwise, the operation is noncommutative. ...

A or BB or A

Bitwise operation

Disjunction is often used for bitwise operations. Examples:

  • 0 or 0 = 0
  • 0 or 1 = 1
  • 1 or 0 = 1
  • 1 or 1 = 1
  • 1010 or 1110 = 1110

Note that in computer science the OR operator can be used to set a bit to 1 by OR-ing the bit with 1. This article is about the unit of information. ...


Union

The union used in set theory is defined in terms of a logical disjunction: xAB if and only if (xA) ∨ (xB). Because of this, logical disjunction satisfies many of the same identities as set-theoretic union, such as associativity, commutativity, distributivity, and de Morgan's laws. In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... note that demorgans laws are also a big part in circut design. ...


Notes

  • Boole, closely following analogy with ordinary mathematics, premised, as a necessary condition to the definition of "x + y", that x and y were mutually exclusive. Jevons, and practically all mathematical logicians after him, advocated, on various grounds, the definition of "logical addition" in a form which does not necessitate mutual exclusiveness.

This article is not about George Boolos, another mathematical logician. ... William Stanley Jevons (September 1, 1835 - August 13, 1882), English economist and logician, was born in Liverpool. ...

See also

Logical operators

Exclusive disjunction (usual symbol xor) is a logical operator that results in true if one of the operands (not both) is true. ... AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ... XNOR Logic Gate Symbol Logical equality is a logical operator that corresponds to equality in boolean algebra and to the logical biconditional in propositional calculus. ... In logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... NAND Logic gate The Sheffer stroke, written | or ↑, denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as not both. It is also called the alternative denial, since it says in effect that at least one of its operands is false. ... NOR Logic Gate The logical NOR or joint denial is a boolean logic operator which produces a result that is the inverse of logical or. ... Negation (i. ...

Related topics

The logical fallacy of affirming a disjunct occurs in a disjunctive syllogism when an argument takes the form: Either A or B (this is the disjunct) A (Affirming the middle term) Therefore, not B The fallacy lies in concluding that B must be false because A is true; in fact... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... Algebra of sets Ampheck Boole, George Boolean algebra Boolean domain Boolean function Boolean logic Boolean implicant Boolean prime ideal theorem Boolean-valued function Boolean-valued model Boolean satisfiability problem Booles syllogistic Canonical form (Boolean algebra) Characteristic function Compactness theorem Complete Boolean algebra De Morgan, Augustus De Morgans laws... A boolean domain B is a generic 2-element set, say, B = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true. ... In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ... Boolean logic is a complete system for logical operations. ... A boolean-valued function is a function of the type , where is an arbitrary set, where is a generic 2-element set, typically , and where the latter is frequently interpreted for logical applications as . ... A disjunctive syllogism, also known as modus tollendo ponens (literally: mode which, by denying, affirms) is a valid, simple argument form: P or Q Not P Therefore, Q In logical operator notation: ¬ where represents the logical assertion. ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ... A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. ... In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ... In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ... Zeroth-order logic is a term in popular use among practitioners for the subject matter otherwise known as boolean functions, monadic predicate logic, propositional calculus, or sentential calculus. ...

External links


  Results from FactBites:
 
Logical disjunction - Wikipedia, the free encyclopedia (459 words)
In mathematics, logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true.
In logic and mathematics, a disjunction is an "or statement".
Because of this, logical disjunction satisfies many of the same identities as set-theoretic union, such as associativity, commutativity, distributivity, and de Morgan's laws.
Logical NOR - Wikipedia, the free encyclopedia (371 words)
The Logical nor or joint denial is a boolean logic operator which produces a result that is the inverse of logical or.
The two-input logical NOR operator is commonly described by a truth table, describing the output state for all possible input combinations:
NOR has the interesting feature that all other logical operators can be expressed by various functions of nor.
  More results at FactBites »

 
 

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