reconoce al tipo de conectiva que indica que dos proposiciones son verdaderas al mismo tiempo.

obtenga reconoce al tipo de conectiva que indica que dos proposiciones son verdaderas al mismo tiempo. de este sitio.

Reconoce al tipo de conectiva que indica que dos proposiciones son verdaderas al mismo tiempo. A)

La conjunción es una proposición compuesta, resultante de la combinación de dos proposiciones simples con la palabra “y”.La conjunción de dos proposiciones simp…

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Reconoce al tipo de conectiva que indica que dos proposiciones son verdaderas al mismo tiempo.

A) - Brainly.lat

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Lógica proposicional 3

Se llama proposición a todo enunciado respecto del cual se disponga de un criterio que nos permita afirmar que su contenido es verdadero (V) o falso (F).

Lógica proposicional 3

Concepto de proposiciónOperaciones con proposiciones

Concepto de proposición

Se llama proposición a todo enunciado respecto del cual se disponga de un criterio que nos permita afirmar que su contenido es verdadero (V) o falso (F).

A toda proposición se le puede asignar uno de los dos valores: verdadero (V) o falso (F), pero nunca ambos a la vez.

Son ejemplos de proposiciones: En matemáticas, algunas proposiciones reciben nombres especiales.

Destacaremos, como más importantes, los llamados postulados o axiomas (palabras en la actualidad sinónimas en algunos casos) y los teoremas. Postulados o Axiomas: Son aquellas proposiciones que se dan como ciertas desde un principio: Ejemplo: "Por dos puntos distintos pasa siempre una recta y sólo una". Teoremas: Son aquellas proposiciones cuya certeza es demostrada por un razonamiento, conforme a las leyes de la lógica.

Carmen es prima de José. (Este enunciado puede ser, en efecto, verdadero o falso.)

Sofía está cansada. (Como en el ejemplo anterior, este enunciado puede ser verdadero o falso.)

Los niños necesitan jugar.

Sócrates es un hombre.

La cultura es fundamental para la humanidad.

x + 4 = 26 (Nótese que según el valor que adopte la variable x, este enunciado puede ser verdadero o falso).

Por el contrario, no serían proposiciones los siguientes enunciados:

¡Dile que pase!

¡Pon el libro en la estantería!

Todas las personas tienen que cuidarse.

¿Será eso cierto?

En efecto, los ejemplos g) y h) no constituyen proposiciones, puesto que los enunciados que en ellos figuran no son susceptibles de adaptar uno de los valores: verdadero o falso, y puede afirmarse, en general, que ni las frases imperativas, como g) y h), ni las frases interrogativas, del tipo j), constituyen proposiciones.

En cuanto al ejemplo i), tampoco constituye una proposición, puesto que la partícula "todas" indica que el enunciado debe cumplirse en todos los casos, y por tanto ser siempre verdadero, con lo que desaparece el carácter de bivalencia que caracteriza a toda proposición. "El triángulo abc es equilátero", es una proposición. Se trata de una sentencia de la que se puede afirmar si es verdadera o falsa.

De haber suprimido esta palabra, el enunciado resultante "las personas tienen que cuidarse", sí constituiría una proposición, ya que dicho enunciado podría ser verdadero o falso.

Proposición 1. ¿A qué se llama proposición? 2. ¿Cómo se designa cada una de las proposiciones simples?

Operaciones con proposiciones   

En la práctica, es poco frecuente encontrar enunciados constituidos por una única proposición, como las de los ejemplos anteriores [a) y e)]. Lo normal es que aparezcan enunciados formados por varias proposiciones elementales enlazadas mediante partículas gramaticales, tales como "y", "o", "si", "no"…, resultando de este modo proposiciones compuestas, formadas por una o varias proposiciones simples enlazadas mediante partículas gramaticales.

"El equipo jugó con orden y disciplina pese a que actuó con varios suplentes", es una proposición compuesta: se ha obtenido a partir de las proposiciones:

a1) el equipo jugó con orden; a2) el equipo jugó con disciplina; a3) el equipo actuó con varios suplentes.

Las proposiciones a1 y a2 se han unido mediante la conjunción "y"; a2 y a3 por medio de la expresión "pese a que".

Para la representación abreviada de las proposiciones compuestas, y ya que interesa la estructura de la proposición, es decir, la configuración de una proposición a partir de varias proposiciones simples, más que el significado concreto de la proposición, pueden utilizarse las siguientes reglas:

Se designa cada una de las proposiciones simples mediante una letra mayúscula del alfabeto.Ejemplos: Luis viene pronto = PMe marché = Q

Las partículas gramaticales, también denominadas por su carácter de elemento enlazante, conectores lógicos o conectivos proposicionales, se representan mediante símbolos estandarizados, que son los siguientes:

a) "No" (que indica negación): ? , -, N, ~. Ejemplo: La negación de la proposición "Juan quiere correr", sería "Juan no quiere correr". Si se designa por P la proposición original, su negación puede expresarse mediante ?P, -P, NP, ~ P.b) "Y" (que indica conjunción): ? Ejemplo: "El viento sopla y caen las hojas". Es fácil observar que la partícula "y" indica que se dan ambos sucesos a la vez. Su representación sería: P??Q. Denominando P a la proposición: "el viento sopla" y Q a la proposición: "caen las hojas". La noción de conjunción es análoga, en el campo de las proposiciones, a la intersección en teoría de conjuntos. c) "O" (que indica disyunción no exclusiva): ?. El término "disyunción" implica que puede verificarse una u otra de las dos proposiciones elementales que forman la proposición. (En adelante, la palabra proposición hará referencia a una proposición compuesta, ya que es el caso más usual, reservándose el nombre de proposiciones elementales para cada una de las que integran la proposición.) El carácter de "no exclusiva" indica que pueden verificarse las dos proposiciones a la vez, es decir, que una de las proposiciones no excluye a la otra. Ejemplo: Se considera la proposición "iremos a pasear o cantaremos"; se comprende fácilmente que son posibles tres opciones: 1. Ir a pasear. 2. Cantar.    3. Ir a pasear y cantar al mismo tiempo.

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Bicondicional

Bicondicional

Ir a la navegación Ir a la búsqueda Bicondicional

Diagrama de Venn de la conectiva

Nomenclatura

Lenguaje natural A si y solo si B

A es equivalente a B

Lenguaje formal

{\displaystyle A\leftrightarrow B}

Operador booleano

{\displaystyle \leftrightarrow \ \iff \ =}

Operador de conjuntos

{\displaystyle \cup }

Puerta lógica Tabla de verdad

{\displaystyle {\begin{array}{c|c||c}A&B&A\leftrightarrow B\\\hline V&V&V\\V&F&F\\F&V&F\\F&F&V\\\end{array}}}

[editar datos en Wikidata]

Conectivas lógicas Tautología

{\displaystyle \top }

Conjunción opuesta

{\displaystyle \uparrow }

Implicación opuesta

{\displaystyle \leftarrow }

Condicional material

{\displaystyle \rightarrow }

Disyunción lógica

{\displaystyle \lor }

Negación lógica

{\displaystyle \lnot }

Disyunción exclusiva

{\displaystyle \nleftrightarrow }

Bicondicional

{\displaystyle \leftrightarrow }

Afirmación lógica Disyunción opuesta

{\displaystyle \downarrow }

Adjunción lógica

{\displaystyle \nrightarrow }

Adjunción opuesta

{\displaystyle \nleftarrow }

Conjunción lógica

{\displaystyle \land }

Contradicción

{\displaystyle \bot }

vte

En algunos contextos en matemáticas y lógica, un bicondicional (equivalencia o doble implicación, en ocasiones abreviado en español como sii) es un operador lógico binario, es decir, una función

{\displaystyle \leftrightarrow :B\times B\rightarrow B}

, siendo B cualquier conjunto con

{\displaystyle |B|=2}

, aunque es común que se considere a B como

{\displaystyle B=\{V,F\}}

o

{\displaystyle B=\{0,1\}}

. El bicondicional también se desempeña como conectivo lógico, permitiendo formular expresiones de la forma «P si y solo si Q», que es verdadera en el caso de que ambos componentes tengan el mismo valor de verdad. En otro contexto el bicondicional representa la equivalencia lógica entre dos proposiciones.

Índice

1 Definición

2 Representación y lectura

3 Ejemplos 4 Referencias

Definición[editar]

El valor de verdad de un bicondicional « si y solo si » es verdadero cuando ambas proposiciones (p y q) tienen el mismo valor de verdad, es decir, ambas son verdaderas o falsas simultáneamente; de lo contrario, es falso.

Se tiene así que la afirmación « si y solo si » es lógicamente equivalente al par de afirmaciones «Si , entonces », y «si , entonces ». Escrito utilizando conectivas lógicas :

{\displaystyle p\leftrightarrow q\equiv (p\to q)\wedge (q\to p)}

.

De manera más precisa, el operador bicondicional tiene la siguiente tabla de verdad:1​2​

si y solo si

p q

p ↔ q

V V V V F F F V F F F V

Representación y lectura[editar]

Una forma de expresar el bicondicional es decir que Q es una condición necesaria y suficiente para P. También se conoce con el nombre de coimplicación.3​

En español se usan las abreviaturas sii, ssi y syss, de modo que es equivalente a “ sii ”. En inglés se abrevia iff (If and only if).

En Lógica y en matemáticas los símbolos empleados para denotar el bicondicional son

{\displaystyle \leftrightarrow }

,

{\displaystyle \iff }

y ≡. La notacion

{\displaystyle \leftrightarrow }

se utiliza frecuentemente como un conectivo u operador lógico, que permite combinar dos proposiciones más simples para generar una proposición compuesta de la forma

{\displaystyle P\leftrightarrow Q}

, mientras que la segunda y tercera notación se emplean casi siempre para denotar la relación de equivalencia lógica entre dos proposiciones lógicas. El significado de cada notación depende fuertemente del contexto en que se utilicen.4​ 5​

Adicionalmente, en el ámbito de la lógica digital, el funcionamiento del operador bicondicional puede emularse mediante la puerta lógica XNOR, y a la negación de la puerta XOR.

Ejemplos[editar]

«

{\displaystyle 2<10\leftrightarrow 5|20}

» y «

{\displaystyle 5>9\leftrightarrow {\sqrt {17}}<{\sqrt[{3}]{6}}}

» son bicondicionales verdaderos.

{\displaystyle c={\text{mcm}}(a,b)\iff c\mathbb {Z} =a\mathbb {Z} \ \cap \ b\mathbb {Z} }

, donde

{\displaystyle n\mathbb {Z} }

denota a los múltiplos enteros de n.

Es esencial distinguir entre las relaciones bicondicionales y las que son meramente condicionales.

Por ejemplo, nótese la diferencia entre las dos proposiciones siguientes:

Una persona es mayor de edad si posee legalmente el carné de conductor.

O bien,

Una persona es mayor de edad si y solo si posee legalmente el carné de conductor.

La primera proposición es correcta, puesto que es imposible poseer legalmente el carné de conducir siendo menor de edad. Por tanto, si se tiene el carné, se tiene que ser obligatoriamente mayor de edad.6​

La segunda es incorrecta, puesto que la relación entre "tener el carné de conducir" y "ser mayor de edad" no es bicondicional. Dicho de otro modo: se puede ser mayor de edad sin tener el carné de conducir.7​

fuente : es.wikipedia.org