Sr Examen
Lógica matemática/
(a∨b)∨(a∧c)=a∨b∧a∨c=a∧b∧(a∨c)=ab∧(a∨c)(ab∨abc∨bc∨c)(c∨ac∨abc)=(ab∨c)(ac∨c)=abac∨cac∨abc∨cc
Expresión (a∨b)∨(a∧c)=a∨b∧a∨c=a∧b∧(a∨c)=ab∧(a∨c)(ab∨abc∨bc∨c)(c∨ac∨abc)=(ab∨c)(ac∨c)=abac∨cac∨abc∨cc
El profesor se sorprenderá mucho al ver tu solución correcta😉
⌨
Solución
Ha introducido
[src]
(a∧b∧(a∨c))⇔(a∨b∨(a∧c))⇔(a∨c∨(a∧b))⇔(c∨(a∧c)∨(a∧b∧c))⇔((c∨(a∧b))∧(c∨(a∧c)))⇔(a∧b∧(a∨c)∧(c∨(a∧c)∨(a∧b∧c))∧(c∨(a∧b)∨(b∧c)∨(a∧b∧c)))
$$\left(\left(c \vee \left(a \wedge b\right)\right) \wedge \left(c \vee \left(a \wedge c\right)\right)\right) ⇔ \left(a \wedge b \wedge \left(a \vee c\right)\right) ⇔ \left(a \wedge b \wedge \left(a \vee c\right) \wedge \left(c \vee \left(a \wedge c\right) \vee \left(a \wedge b \wedge c\right)\right) \wedge \left(c \vee \left(a \wedge b\right) \vee \left(b \wedge c\right) \vee \left(a \wedge b \wedge c\right)\right)\right) ⇔ \left(a \vee b \vee \left(a \wedge c\right)\right) ⇔ \left(a \vee c \vee \left(a \wedge b\right)\right) ⇔ \left(c \vee \left(a \wedge c\right) \vee \left(a \wedge b \wedge c\right)\right)$$
Solución detallada
$$a \wedge b \wedge \left(a \vee c\right) = a \wedge b$$
$$a \vee b \vee \left(a \wedge c\right) = a \vee b$$
$$a \vee c \vee \left(a \wedge b\right) = a \vee c$$
$$c \vee \left(a \wedge c\right) \vee \left(a \wedge b \wedge c\right) = c$$
$$c \vee \left(a \wedge c\right) = c$$
$$\left(c \vee \left(a \wedge b\right)\right) \wedge \left(c \vee \left(a \wedge c\right)\right) = c$$
$$c \vee \left(a \wedge b\right) \vee \left(b \wedge c\right) \vee \left(a \wedge b \wedge c\right) = c \vee \left(a \wedge b\right)$$
$$a \wedge b \wedge \left(a \vee c\right) \wedge \left(c \vee \left(a \wedge c\right) \vee \left(a \wedge b \wedge c\right)\right) \wedge \left(c \vee \left(a \wedge b\right) \vee \left(b \wedge c\right) \vee \left(a \wedge b \wedge c\right)\right) = a \wedge b \wedge c$$
$$\left(\left(c \vee \left(a \wedge b\right)\right) \wedge \left(c \vee \left(a \wedge c\right)\right)\right) ⇔ \left(a \wedge b \wedge \left(a \vee c\right)\right) ⇔ \left(a \wedge b \wedge \left(a \vee c\right) \wedge \left(c \vee \left(a \wedge c\right) \vee \left(a \wedge b \wedge c\right)\right) \wedge \left(c \vee \left(a \wedge b\right) \vee \left(b \wedge c\right) \vee \left(a \wedge b \wedge c\right)\right)\right) ⇔ \left(a \vee b \vee \left(a \wedge c\right)\right) ⇔ \left(a \vee c \vee \left(a \wedge b\right)\right) ⇔ \left(c \vee \left(a \wedge c\right) \vee \left(a \wedge b \wedge c\right)\right) = \left(a \vee \neg b\right) \wedge \left(a \vee \neg c\right) \wedge \left(b \vee \neg a\right) \wedge \left(b \vee \neg c\right) \wedge \left(c \vee \neg a\right) \wedge \left(c \vee \neg b\right)$$
$$a \vee b \vee \left(a \wedge c\right) = a \vee b$$
$$a \vee c \vee \left(a \wedge b\right) = a \vee c$$
$$c \vee \left(a \wedge c\right) \vee \left(a \wedge b \wedge c\right) = c$$
$$c \vee \left(a \wedge c\right) = c$$
$$\left(c \vee \left(a \wedge b\right)\right) \wedge \left(c \vee \left(a \wedge c\right)\right) = c$$
$$c \vee \left(a \wedge b\right) \vee \left(b \wedge c\right) \vee \left(a \wedge b \wedge c\right) = c \vee \left(a \wedge b\right)$$
$$a \wedge b \wedge \left(a \vee c\right) \wedge \left(c \vee \left(a \wedge c\right) \vee \left(a \wedge b \wedge c\right)\right) \wedge \left(c \vee \left(a \wedge b\right) \vee \left(b \wedge c\right) \vee \left(a \wedge b \wedge c\right)\right) = a \wedge b \wedge c$$
$$\left(\left(c \vee \left(a \wedge b\right)\right) \wedge \left(c \vee \left(a \wedge c\right)\right)\right) ⇔ \left(a \wedge b \wedge \left(a \vee c\right)\right) ⇔ \left(a \wedge b \wedge \left(a \vee c\right) \wedge \left(c \vee \left(a \wedge c\right) \vee \left(a \wedge b \wedge c\right)\right) \wedge \left(c \vee \left(a \wedge b\right) \vee \left(b \wedge c\right) \vee \left(a \wedge b \wedge c\right)\right)\right) ⇔ \left(a \vee b \vee \left(a \wedge c\right)\right) ⇔ \left(a \vee c \vee \left(a \wedge b\right)\right) ⇔ \left(c \vee \left(a \wedge c\right) \vee \left(a \wedge b \wedge c\right)\right) = \left(a \vee \neg b\right) \wedge \left(a \vee \neg c\right) \wedge \left(b \vee \neg a\right) \wedge \left(b \vee \neg c\right) \wedge \left(c \vee \neg a\right) \wedge \left(c \vee \neg b\right)$$
Simplificación
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$$\left(a \vee \neg b\right) \wedge \left(a \vee \neg c\right) \wedge \left(b \vee \neg a\right) \wedge \left(b \vee \neg c\right) \wedge \left(c \vee \neg a\right) \wedge \left(c \vee \neg b\right)$$
(a∨(¬b))∧(a∨(¬c))∧(b∨(¬a))∧(b∨(¬c))∧(c∨(¬a))∧(c∨(¬b))
Tabla de verdad
+---+---+---+--------+ | a | b | c | result | +===+===+===+========+ | 0 | 0 | 0 | 1 | +---+---+---+--------+ | 0 | 0 | 1 | 0 | +---+---+---+--------+ | 0 | 1 | 0 | 0 | +---+---+---+--------+ | 0 | 1 | 1 | 0 | +---+---+---+--------+ | 1 | 0 | 0 | 0 | +---+---+---+--------+ | 1 | 0 | 1 | 0 | +---+---+---+--------+ | 1 | 1 | 0 | 0 | +---+---+---+--------+ | 1 | 1 | 1 | 1 | +---+---+---+--------+
FNC
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Ya está reducido a FNC
$$\left(a \vee \neg b\right) \wedge \left(a \vee \neg c\right) \wedge \left(b \vee \neg a\right) \wedge \left(b \vee \neg c\right) \wedge \left(c \vee \neg a\right) \wedge \left(c \vee \neg b\right)$$
(a∨(¬b))∧(a∨(¬c))∧(b∨(¬a))∧(b∨(¬c))∧(c∨(¬a))∧(c∨(¬b))
FNDP
[src]
$$\left(a \wedge b \wedge c\right) \vee \left(\neg a \wedge \neg b \wedge \neg c\right)$$
(a∧b∧c)∨((¬a)∧(¬b)∧(¬c))
FND
[src]
$$\left(a \wedge b \wedge c\right) \vee \left(\neg a \wedge \neg b \wedge \neg c\right) \vee \left(a \wedge b \wedge c \wedge \neg a\right) \vee \left(a \wedge b \wedge c \wedge \neg b\right) \vee \left(a \wedge b \wedge c \wedge \neg c\right) \vee \left(a \wedge b \wedge \neg a \wedge \neg b\right) \vee \left(a \wedge c \wedge \neg a \wedge \neg c\right) \vee \left(a \wedge \neg a \wedge \neg b \wedge \neg c\right) \vee \left(b \wedge c \wedge \neg b \wedge \neg c\right) \vee \left(b \wedge \neg a \wedge \neg b \wedge \neg c\right) \vee \left(c \wedge \neg a \wedge \neg b \wedge \neg c\right) \vee \left(a \wedge b \wedge c \wedge \neg a \wedge \neg b\right) \vee \left(a \wedge b \wedge c \wedge \neg a \wedge \neg c\right) \vee \left(a \wedge b \wedge c \wedge \neg b \wedge \neg c\right) \vee \left(a \wedge b \wedge \neg a \wedge \neg b \wedge \neg c\right) \vee \left(a \wedge c \wedge \neg a \wedge \neg b \wedge \neg c\right) \vee \left(b \wedge c \wedge \neg a \wedge \neg b \wedge \neg c\right) \vee \left(a \wedge b \wedge c \wedge \neg a \wedge \neg b \wedge \neg c\right)$$
(a∧b∧c)∨(a∧b∧c∧(¬a))∨(a∧b∧c∧(¬b))∨(a∧b∧c∧(¬c))∨((¬a)∧(¬b)∧(¬c))∨(a∧b∧(¬a)∧(¬b))∨(a∧c∧(¬a)∧(¬c))∨(b∧c∧(¬b)∧(¬c))∨(a∧(¬a)∧(¬b)∧(¬c))∨(b∧(¬a)∧(¬b)∧(¬c))∨(c∧(¬a)∧(¬b)∧(¬c))∨(a∧b∧c∧(¬a)∧(¬b))∨(a∧b∧c∧(¬a)∧(¬c))∨(a∧b∧c∧(¬b)∧(¬c))∨(a∧b∧(¬a)∧(¬b)∧(¬c))∨(a∧c∧(¬a)∧(¬b)∧(¬c))∨(b∧c∧(¬a)∧(¬b)∧(¬c))∨(a∧b∧c∧(¬a)∧(¬b)∧(¬c))
FNCD
[src]
$$\left(a \vee \neg b\right) \wedge \left(a \vee \neg c\right) \wedge \left(b \vee \neg a\right) \wedge \left(b \vee \neg c\right) \wedge \left(c \vee \neg a\right) \wedge \left(c \vee \neg b\right)$$
(a∨(¬b))∧(a∨(¬c))∧(b∨(¬a))∧(b∨(¬c))∧(c∨(¬a))∧(c∨(¬b))