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