Sr Examen
Expresión not(x1ornot(x1)andx2)ornot(not(x1andx2)andx3orx1andx2andnot(x3)andx4)
El profesor se sorprenderá mucho al ver tu solución correcta😉
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Solución
Ha introducido
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(¬(x1∨(x2∧(¬x1))))∨(¬((x3∧(¬(x1∧x2)))∨(x1∧x2∧x4∧(¬x3))))
$$\neg \left(x_{1} \vee \left(x_{2} \wedge \neg x_{1}\right)\right) \vee \neg \left(\left(x_{3} \wedge \neg \left(x_{1} \wedge x_{2}\right)\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{4} \wedge \neg x_{3}\right)\right)$$
Solución detallada
$$x_{1} \vee \left(x_{2} \wedge \neg x_{1}\right) = x_{1} \vee x_{2}$$
$$\neg \left(x_{1} \vee \left(x_{2} \wedge \neg x_{1}\right)\right) = \neg x_{1} \wedge \neg x_{2}$$
$$\neg \left(x_{1} \wedge x_{2}\right) = \neg x_{1} \vee \neg x_{2}$$
$$x_{3} \wedge \neg \left(x_{1} \wedge x_{2}\right) = x_{3} \wedge \left(\neg x_{1} \vee \neg x_{2}\right)$$
$$\left(x_{3} \wedge \neg \left(x_{1} \wedge x_{2}\right)\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{4} \wedge \neg x_{3}\right) = \left(x_{1} \vee x_{3}\right) \wedge \left(x_{2} \vee x_{3}\right) \wedge \left(x_{3} \vee x_{4}\right) \wedge \left(\neg x_{1} \vee \neg x_{2} \vee \neg x_{3}\right)$$
$$\neg \left(\left(x_{3} \wedge \neg \left(x_{1} \wedge x_{2}\right)\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{4} \wedge \neg x_{3}\right)\right) = \left(\neg x_{1} \wedge \neg x_{3}\right) \vee \left(\neg x_{2} \wedge \neg x_{3}\right) \vee \left(\neg x_{3} \wedge \neg x_{4}\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{3}\right)$$
$$\neg \left(x_{1} \vee \left(x_{2} \wedge \neg x_{1}\right)\right) \vee \neg \left(\left(x_{3} \wedge \neg \left(x_{1} \wedge x_{2}\right)\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{4} \wedge \neg x_{3}\right)\right) = \left(\neg x_{1} \wedge \neg x_{2}\right) \vee \left(\neg x_{1} \wedge \neg x_{3}\right) \vee \left(\neg x_{2} \wedge \neg x_{3}\right) \vee \left(\neg x_{3} \wedge \neg x_{4}\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{3}\right)$$
$$\neg \left(x_{1} \vee \left(x_{2} \wedge \neg x_{1}\right)\right) = \neg x_{1} \wedge \neg x_{2}$$
$$\neg \left(x_{1} \wedge x_{2}\right) = \neg x_{1} \vee \neg x_{2}$$
$$x_{3} \wedge \neg \left(x_{1} \wedge x_{2}\right) = x_{3} \wedge \left(\neg x_{1} \vee \neg x_{2}\right)$$
$$\left(x_{3} \wedge \neg \left(x_{1} \wedge x_{2}\right)\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{4} \wedge \neg x_{3}\right) = \left(x_{1} \vee x_{3}\right) \wedge \left(x_{2} \vee x_{3}\right) \wedge \left(x_{3} \vee x_{4}\right) \wedge \left(\neg x_{1} \vee \neg x_{2} \vee \neg x_{3}\right)$$
$$\neg \left(\left(x_{3} \wedge \neg \left(x_{1} \wedge x_{2}\right)\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{4} \wedge \neg x_{3}\right)\right) = \left(\neg x_{1} \wedge \neg x_{3}\right) \vee \left(\neg x_{2} \wedge \neg x_{3}\right) \vee \left(\neg x_{3} \wedge \neg x_{4}\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{3}\right)$$
$$\neg \left(x_{1} \vee \left(x_{2} \wedge \neg x_{1}\right)\right) \vee \neg \left(\left(x_{3} \wedge \neg \left(x_{1} \wedge x_{2}\right)\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{4} \wedge \neg x_{3}\right)\right) = \left(\neg x_{1} \wedge \neg x_{2}\right) \vee \left(\neg x_{1} \wedge \neg x_{3}\right) \vee \left(\neg x_{2} \wedge \neg x_{3}\right) \vee \left(\neg x_{3} \wedge \neg x_{4}\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{3}\right)$$
Simplificación
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$$\left(\neg x_{1} \wedge \neg x_{2}\right) \vee \left(\neg x_{1} \wedge \neg x_{3}\right) \vee \left(\neg x_{2} \wedge \neg x_{3}\right) \vee \left(\neg x_{3} \wedge \neg x_{4}\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{3}\right)$$
(x1∧x2∧x3)∨((¬x1)∧(¬x2))∨((¬x1)∧(¬x3))∨((¬x2)∧(¬x3))∨((¬x3)∧(¬x4))
Tabla de verdad
+----+----+----+----+--------+ | x1 | x2 | x3 | x4 | result | +====+====+====+====+========+ | 0 | 0 | 0 | 0 | 1 | +----+----+----+----+--------+ | 0 | 0 | 0 | 1 | 1 | +----+----+----+----+--------+ | 0 | 0 | 1 | 0 | 1 | +----+----+----+----+--------+ | 0 | 0 | 1 | 1 | 1 | +----+----+----+----+--------+ | 0 | 1 | 0 | 0 | 1 | +----+----+----+----+--------+ | 0 | 1 | 0 | 1 | 1 | +----+----+----+----+--------+ | 0 | 1 | 1 | 0 | 0 | +----+----+----+----+--------+ | 0 | 1 | 1 | 1 | 0 | +----+----+----+----+--------+ | 1 | 0 | 0 | 0 | 1 | +----+----+----+----+--------+ | 1 | 0 | 0 | 1 | 1 | +----+----+----+----+--------+ | 1 | 0 | 1 | 0 | 0 | +----+----+----+----+--------+ | 1 | 0 | 1 | 1 | 0 | +----+----+----+----+--------+ | 1 | 1 | 0 | 0 | 1 | +----+----+----+----+--------+ | 1 | 1 | 0 | 1 | 0 | +----+----+----+----+--------+ | 1 | 1 | 1 | 0 | 1 | +----+----+----+----+--------+ | 1 | 1 | 1 | 1 | 1 | +----+----+----+----+--------+
FNC
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$$\left(x_{1} \vee \neg x_{1} \vee \neg x_{3}\right) \wedge \left(x_{1} \vee \neg x_{2} \vee \neg x_{3}\right) \wedge \left(x_{2} \vee \neg x_{1} \vee \neg x_{3}\right) \wedge \left(x_{2} \vee \neg x_{2} \vee \neg x_{3}\right) \wedge \left(x_{3} \vee \neg x_{1} \vee \neg x_{3}\right) \wedge \left(x_{3} \vee \neg x_{2} \vee \neg x_{3}\right) \wedge \left(x_{1} \vee \neg x_{1} \vee \neg x_{2} \vee \neg x_{3}\right) \wedge \left(x_{1} \vee \neg x_{1} \vee \neg x_{2} \vee \neg x_{4}\right) \wedge \left(x_{1} \vee \neg x_{1} \vee \neg x_{3} \vee \neg x_{4}\right) \wedge \left(x_{1} \vee \neg x_{2} \vee \neg x_{3} \vee \neg x_{4}\right) \wedge \left(x_{2} \vee \neg x_{1} \vee \neg x_{2} \vee \neg x_{3}\right) \wedge \left(x_{2} \vee \neg x_{1} \vee \neg x_{2} \vee \neg x_{4}\right) \wedge \left(x_{2} \vee \neg x_{1} \vee \neg x_{3} \vee \neg x_{4}\right) \wedge \left(x_{2} \vee \neg x_{2} \vee \neg x_{3} \vee \neg x_{4}\right) \wedge \left(x_{3} \vee \neg x_{1} \vee \neg x_{2} \vee \neg x_{3}\right) \wedge \left(x_{3} \vee \neg x_{1} \vee \neg x_{2} \vee \neg x_{4}\right) \wedge \left(x_{3} \vee \neg x_{1} \vee \neg x_{3} \vee \neg x_{4}\right) \wedge \left(x_{3} \vee \neg x_{2} \vee \neg x_{3} \vee \neg x_{4}\right) \wedge \left(x_{1} \vee \neg x_{1} \vee \neg x_{2} \vee \neg x_{3} \vee \neg x_{4}\right) \wedge \left(x_{2} \vee \neg x_{1} \vee \neg x_{2} \vee \neg x_{3} \vee \neg x_{4}\right) \wedge \left(x_{3} \vee \neg x_{1} \vee \neg x_{2} \vee \neg x_{3} \vee \neg x_{4}\right)$$
(x1∨(¬x1)∨(¬x3))∧(x1∨(¬x2)∨(¬x3))∧(x2∨(¬x1)∨(¬x3))∧(x2∨(¬x2)∨(¬x3))∧(x3∨(¬x1)∨(¬x3))∧(x3∨(¬x2)∨(¬x3))∧(x1∨(¬x1)∨(¬x2)∨(¬x3))∧(x1∨(¬x1)∨(¬x2)∨(¬x4))∧(x1∨(¬x1)∨(¬x3)∨(¬x4))∧(x1∨(¬x2)∨(¬x3)∨(¬x4))∧(x2∨(¬x1)∨(¬x2)∨(¬x3))∧(x2∨(¬x1)∨(¬x2)∨(¬x4))∧(x2∨(¬x1)∨(¬x3)∨(¬x4))∧(x2∨(¬x2)∨(¬x3)∨(¬x4))∧(x3∨(¬x1)∨(¬x2)∨(¬x3))∧(x3∨(¬x1)∨(¬x2)∨(¬x4))∧(x3∨(¬x1)∨(¬x3)∨(¬x4))∧(x3∨(¬x2)∨(¬x3)∨(¬x4))∧(x1∨(¬x1)∨(¬x2)∨(¬x3)∨(¬x4))∧(x2∨(¬x1)∨(¬x2)∨(¬x3)∨(¬x4))∧(x3∨(¬x1)∨(¬x2)∨(¬x3)∨(¬x4))
FND
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Ya está reducido a FND
$$\left(\neg x_{1} \wedge \neg x_{2}\right) \vee \left(\neg x_{1} \wedge \neg x_{3}\right) \vee \left(\neg x_{2} \wedge \neg x_{3}\right) \vee \left(\neg x_{3} \wedge \neg x_{4}\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{3}\right)$$
(x1∧x2∧x3)∨((¬x1)∧(¬x2))∨((¬x1)∧(¬x3))∨((¬x2)∧(¬x3))∨((¬x3)∧(¬x4))
FNCD
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$$\left(x_{1} \vee \neg x_{2} \vee \neg x_{3}\right) \wedge \left(x_{2} \vee \neg x_{1} \vee \neg x_{3}\right) \wedge \left(x_{3} \vee \neg x_{1} \vee \neg x_{2} \vee \neg x_{4}\right)$$
(x1∨(¬x2)∨(¬x3))∧(x2∨(¬x1)∨(¬x3))∧(x3∨(¬x1)∨(¬x2)∨(¬x4))
FNDP
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$$\left(\neg x_{1} \wedge \neg x_{2}\right) \vee \left(\neg x_{1} \wedge \neg x_{3}\right) \vee \left(\neg x_{2} \wedge \neg x_{3}\right) \vee \left(\neg x_{3} \wedge \neg x_{4}\right) \vee \left(x_{1} \wedge x_{2} \wedge x_{3}\right)$$
(x1∧x2∧x3)∨((¬x1)∧(¬x2))∨((¬x1)∧(¬x3))∨((¬x2)∧(¬x3))∨((¬x3)∧(¬x4))