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Completar tablas de verdad y demostrar usando el método por contradicción.

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Por favor lea lo siguiente antes de intentar la pregunta:
Si no ha proporcionado una respuesta a cada intervalo de entrada de una pregunta o parte de la pregunta e intenta enviar sus respuestas a la pregunta o la parte, aparecerá el mensaje \"No se puede enviar la respuesta: verifique si hay errores\". En realidad, su respuesta ha sido enviada, pero al sistema solo le preocupa que no haya enviado una respuesta a cada intervalo de entrada. Por este motivo, asegúrese de proporcionar una respuesta a cada brecha de entrada en la pregunta o parte antes de enviarla. Incluso si no está seguro de la respuesta, escriba lo que cree que es más probable que sea correcto; Siempre puedes cambiar tu respuesta o volver a intentar la pregunta.

Al igual que con todas las preguntas, puede haber partes donde puede elegir \"Mostrar pasos\". Esto puede dar una pista, o puede presentar subpartes que le ayudarán a resolver esa parte de la pregunta. Además, recuerde presionar siempre el botón \"Mostrar comentarios\" al final de cada parte. A veces, se proporcionarán comentarios útiles aquí y, a menudo, dependerán de la respuesta correcta y se vincularán a otras partes de la pregunta. Por lo tanto, siempre vuelva a intentar las partes hasta que obtenga las marcas completas, y luego mire la retroalimentación nuevamente.

Tenga en cuenta que para ver los comentarios de una parte en particular de una pregunta, debe proporcionar una respuesta completa (pero no necesariamente correcta) a esa parte. Sin embargo, no se preocupe, ya que puede ver los comentarios y luego enmendar su respuesta en consecuencia.

Además, al igual que con todas las preguntas, la elección de revelar las respuestas solo le mostrará las respuestas que cambian cada vez que se carga la pregunta (es decir, las respuestas a las preguntas aleatorias); Las respuestas fijas no serán reveladas.

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Complete la siguiente tabla de verdad con las proposiciones $ p $ y $ q $.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$p$	$q$	$p \\Rightarrow q$	$q \\Rightarrow p$	$\\neg p$	$\\neg q$	$( \\neg p ) \\Rightarrow q$	$( \\neg q ) \\Rightarrow ( \\neg p )$	$\\neg ( p \\Rightarrow q )$	$p \\land ( \\neg q )$	$( \\neg p ) \\land ( \\neg q )$
V	V	[[0]]	[[2]]	[[21]]	[[23]]	[[9]]	[[12]]	[[27]]	[[30]]	[[33]]
V	F	[[19]]	[[3]]	[[22]]	[[7]]	[[10]]	[[26]]	[[15]]	[[16]]	[[34]]
F	V	[[1]]	[[20]]	[[5]]	[[24]]	[[11]]	[[13]]	[[28]]	[[31]]	[[35]]
F	F	[[18]]	[[4]]	[[6]]	[[8]]	[[25]]	[[14]]	[[29]]	[[32]]	[[17]]

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Sugerencia: consulte los ejemplos de la tabla de verdad en el Capítulo 2 para obtener orientación

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i) Suponga que $( \\neg q ) \\Rightarrow ( \\neg p )$ es verdadero. ¿Cuál es el valor de verdad de $ p \\Rightarrow q $?? [[0]]

ii) Suponga que $( \\neg q ) \\Rightarrow ( \\neg p )$ es falso. ¿Cuál es el valor de verdad de $ p \\Rightarrow q $ [[1]]

iii) Por lo tanto, podemos decir que \"$( \\neg q ) \\Rightarrow ( \\neg p )$\" and \"$p \\Rightarrow q$\" son lógicamente equivalentes? [[2]]

Sugerencia para la parte i: En la tabla de verdad de la parte a, encuentre todas las filas que tienen una \"V\" en la columna de $ (\\neg q) \\Rightarrow (\\neg p) $. Para cada fila, muévase a lo largo de la fila hasta que llegue a la entrada en la columna para $ p \\Rightarrow q $. Tome nota de este valor de verdad. ¿Qué notas cada vez?

Sugerencia para la parte ii: En la tabla de verdad de la parte a, encuentre todas las filas que tienen una \"F\" en la columna de $ (\\neg q) \\Rightarrow (\\neg p) $. Para cada fila, muévase a lo largo de la fila hasta que llegue a la entrada en la columna para $ p \\Rightarrow q $. Tome nota de este valor de verdad. ¿Qué notas cada vez?

Sugerencia para la parte iii: la definición de \"lógicamente equivalente\" se describe a continuación. Supongamos que tenemos dos propuestas $A$ y $B$, que se acumulan a partir de las dos propuestas $p$ y $q$ (por ejemplo, $A$ podría ser $p \\Rightarrow q$). $A$ y $B$ son lógicamente equivalentes si, por cada par de valores de verdad para $p$ y $q$, el valor de verdad para $A$ y el valor de verdad de $B$ son los mismos.

Falso

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Falso

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Suponga que $(\\neg p) \\Rightarrow q$ es verdadero y $q$ es falso. ¿Cuál es el valor de verdad de $p$?? [[0]].

En la tabla de verdad de la parte a, encuentre la fila donde el valor de verdad de $ (\\neg p) \\Rightarrow q $ es verdadero y el valor de verdad de $q$ es falso. En esta fila, ¿cuál es el valor de verdad de $p$?

Falso

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En esta parte, y las partes e y f, probaremos, por contraposición, la afirmación \"si $ m^2 $ es impar, entonces $m$ es impar\".

¿Cuál es la afirmación contrapositiva de \"si $ m^2 $ es impar, entonces $m$ es impar\"?
[[0]]

Sugerencia: recuerde que la contrapositiva de $p \\Rightarrow q $ (es decir, \"si $ p $ entonces $ q $\") es $ (\\neg q) \\Rightarrow (\\neg p) $ (es decir, \"if $ \\neg q $ entonces $ \\neg p $ \").

Recuerde también que $ \\neg p $ es la negación de $p$. Por ejemplo, la negación de \"$1 <2 $\" es \"$1 \\geq 2 $\"; la negación de \"$5 $ es impar\" es \"$ 5 $ es par\"; La negación de \"Tengo 18 años\" es \"No tengo 18 años\".

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Si $ m^2 $ es impar, entonces $m$ es impar

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Si $ m^2 $ es par, entonces $m$ es par

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Si $ m^2 $ es par, entonces $m$ es impar

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Si $m$ es impar, entonces $ m^2 $ es impar

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Si $m$ es impar, entonces $ m^2 $ es par

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Si $m$ es par, entonces $ m^2 $ es impar

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Si $m$ es par, entonces $ m^2 $ es par

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Por lo tanto, es suficiente probar la afirmación \"si $ m $ es par, entonces $ m ^ 2 $ es par\". Como $ m $ es par, sabemos que:
[[0]]

Sugerencia: vea el capítulo 2 de las notas de la conferencia, donde se utiliza la definición de \"par\" y \"impar\" en algunos ejemplos.

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Existe un entero $ k $ tal que $ m = 4k + 2 $

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Existe un entero $ k $ tal que $ m = 4k $

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Existe un entero $ k $ tal que $ m = 2k $

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Existe un entero $ k $ tal que $ m = 4k + 1 $

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Entonces, ahora podemos escribir $m^2 = 2($[[0]]$)$, y así $m^2$ es par.

(Al ingresar una expresión en el espacio anterior, use * para la multiplicación. Por ejemplo, escriba 10 * k en lugar de 10k, ya que el sistema no siempre puede interpretar esto correctamente).

Necesitamos escribir $m^2$ en la forma $2r$ donde $ r $ es un número entero, para mostrar que $ m^2 $ es par. Dado lo que deducimos sobre $n$ en la parte e, $r$ dependerá de $k$.

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En esta parte de la pregunta, y en las partes h y i, probaremos, por contradicción, la afirmación \"si $ m $ es par, entonces $ m ^ 2 $ es par\". De hecho, es más fácil probar esto directamente, pero de todos modos usaremos una \"prueba por contradicción\", para la práctica.

Es útil usar la letra $ s $ para representar la proposición \"$ m $ es par\", y la letra $ t $ para representar la declaración \"$ m ^ 2 $ es par\".

Entonces, lo que deseamos probar es la sentencia \"$ s \\Rightarrow t $\".

Asumiremos que la negación de la afirmación es verdadera, y luego llegaremos mediante un razonamiento matemático válido a una contradicción. Esto nos dirá que la negación de la sentencia, de hecho, falsa y, por lo tanto, la afirmación en sí es cierta.

i)Ahora, la negación de \"$ s \\Rightarrow t $\" (es decir, \"$ \\neg (s \\Rightarrow t) $\") es lógicamente equivalente a :

[[0]]

ii) Por lo tanto, la negación de \"si $m$ es par, entonces $ m^2 $ es par\" es
[[1]]

Sugerencia para la parte i: use la tabla de verdad de la parte a, tomando $p$ como $s$ y $q$ para ser $t$.

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$s \\land t$

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$s \\lor t$

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$s \\land ( \\neg t )$

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$s \\lor ( \\neg t )$

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$( \\neg s ) \\land ( \\neg t )$

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$( \\neg s ) \\lor ( \\neg t )$

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$( \\neg s ) \\land t$

", "

$( \\neg s ) \\lor t$

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$ m $ es par, y $ m ^ 2 $ es par

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$m$ es par, y $m^2$ es impar

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$m$ es par, o $m^2 $ es par

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$m$ es par, o $ m^2 $ es impar

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\n\n

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Ahora asumimos que $ m $ es par y $m^2$ es impar.

Como $ m $ es par, podemos escribir $ m=2k $ para algún entero $k$. Por lo tanto, $m^2 = 2($[[0]]$)$, y esto muestra que $m^2$ is [[1]].

Por lo tanto, utilizando toda la información en esta parte de la pregunta, podemos decir que $ m ^ 2 $ es [[2]], es claramente la sentencia [[3]] .

Supongamos que $u$ representa la sentencia \"$m^2 $ es[[2]]\".

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Ambos par e impar

", "

ni par ni impar

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falso

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i) Demostramos por un razonamiento matemático válido que \"$ (\\neg (s \\Rightarrow t)) $ implica $u$\". Por lo tanto, la declaración \"$ (\\neg (s \\Rightarrow t)) \\Rightarrow u $\" es [[0]].

ii) Además, se estable que $u$ es [[1]].

iii) Esto implica que la declaración \"$\\neg ( s \\Rightarrow t )$\" es una sentencia [[2]] . (Presiona \"Mostrar paso\" para una indicación).

Indicaciones para la parte iii: mire la tabla de verdad para la parte a. En particular, si \"$ (\\neg p) \\Rightarrow q $\" es verdadero y $ q $ es falso, ¿qué nos dice esto acerca de $ \\neg p $? Ahora tome $ p $ para ser \"$ s \\Rightarrow t $\" y $ q $ para ser $ u $. Teniendo en cuenta lo que escribió para las partes i y ii, y lo que acaba de aprender de la tabla de verdad, ¿qué puede decir sobre la afirmación \"$ \\neg (s \\Rightarrow t) $\"?

verdadero

", "

falso

verdadero

", "

falso

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