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