Sr Examen
Expresión а⇒(b∨c)⇔((a⇒b)|(a⇒c))
El profesor se sorprenderá mucho al ver tu solución correcta😉
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Solución
Ha introducido
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(a⇒(b∨c))⇔((a⇒b)|(a⇒c))
$$\left(a \Rightarrow \left(b \vee c\right)\right) ⇔ \left(\left(a \Rightarrow b\right) | \left(a \Rightarrow c\right)\right)$$
Solución detallada
$$a \Rightarrow \left(b \vee c\right) = b \vee c \vee \neg a$$
$$a \Rightarrow b = b \vee \neg a$$
$$a \Rightarrow c = c \vee \neg a$$
$$\left(a \Rightarrow b\right) | \left(a \Rightarrow c\right) = a \wedge \left(\neg b \vee \neg c\right)$$
$$\left(a \Rightarrow \left(b \vee c\right)\right) ⇔ \left(\left(a \Rightarrow b\right) | \left(a \Rightarrow c\right)\right) = a \wedge \left(b \vee c\right) \wedge \left(\neg b \vee \neg c\right)$$
$$a \Rightarrow b = b \vee \neg a$$
$$a \Rightarrow c = c \vee \neg a$$
$$\left(a \Rightarrow b\right) | \left(a \Rightarrow c\right) = a \wedge \left(\neg b \vee \neg c\right)$$
$$\left(a \Rightarrow \left(b \vee c\right)\right) ⇔ \left(\left(a \Rightarrow b\right) | \left(a \Rightarrow c\right)\right) = a \wedge \left(b \vee c\right) \wedge \left(\neg b \vee \neg c\right)$$
Simplificación
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$$a \wedge \left(b \vee c\right) \wedge \left(\neg b \vee \neg c\right)$$
a∧(b∨c)∧((¬b)∨(¬c))
Tabla de verdad
+---+---+---+--------+ | a | b | c | result | +===+===+===+========+ | 0 | 0 | 0 | 0 | +---+---+---+--------+ | 0 | 0 | 1 | 0 | +---+---+---+--------+ | 0 | 1 | 0 | 0 | +---+---+---+--------+ | 0 | 1 | 1 | 0 | +---+---+---+--------+ | 1 | 0 | 0 | 0 | +---+---+---+--------+ | 1 | 0 | 1 | 1 | +---+---+---+--------+ | 1 | 1 | 0 | 1 | +---+---+---+--------+ | 1 | 1 | 1 | 0 | +---+---+---+--------+
FNC
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Ya está reducido a FNC
$$a \wedge \left(b \vee c\right) \wedge \left(\neg b \vee \neg c\right)$$
a∧(b∨c)∧((¬b)∨(¬c))
FND
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$$\left(a \wedge b \wedge \neg b\right) \vee \left(a \wedge b \wedge \neg c\right) \vee \left(a \wedge c \wedge \neg b\right) \vee \left(a \wedge c \wedge \neg c\right)$$
(a∧b∧(¬b))∨(a∧b∧(¬c))∨(a∧c∧(¬b))∨(a∧c∧(¬c))
FNDP
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$$\left(a \wedge b \wedge \neg c\right) \vee \left(a \wedge c \wedge \neg b\right)$$
(a∧b∧(¬c))∨(a∧c∧(¬b))
FNCD
[src]
$$a \wedge \left(b \vee c\right) \wedge \left(\neg b \vee \neg c\right)$$
a∧(b∨c)∧((¬b)∨(¬c))