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Lógica matemática/ ab⇔ac∨b(a∨¬c⇒ab(¬a∨¬b∨ac))

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
    $$\left(a \wedge c\right) \vee \neg a \vee \neg b = c \vee \neg a \vee \neg b$$
    $$a \wedge b \wedge \left(\left(a \wedge c\right) \vee \neg a \vee \neg b\right) = a \wedge b \wedge c$$
    $$\left(a \vee \neg c\right) \Rightarrow \left(a \wedge b \wedge \left(\left(a \wedge c\right) \vee \neg a \vee \neg b\right)\right) = c \wedge \left(b \vee \neg a\right)$$
    $$b \wedge \left(\left(a \vee \neg c\right) \Rightarrow \left(a \wedge b \wedge \left(\left(a \wedge c\right) \vee \neg a \vee \neg b\right)\right)\right) = b \wedge c$$
    $$\left(a \wedge c\right) \vee \left(b \wedge \left(\left(a \vee \neg c\right) \Rightarrow \left(a \wedge b \wedge \left(\left(a \wedge c\right) \vee \neg a \vee \neg b\right)\right)\right)\right) = c \wedge \left(a \vee b\right)$$
    $$\left(a \wedge b\right) ⇔ \left(\left(a \wedge c\right) \vee \left(b \wedge \left(\left(a \vee \neg c\right) \Rightarrow \left(a \wedge b \wedge \left(\left(a \wedge c\right) \vee \neg a \vee \neg b\right)\right)\right)\right)\right) = \left(\neg a \wedge \neg b\right) \vee \left(\neg a \wedge \neg c\right) \vee \left(\neg b \wedge \neg c\right) \vee \left(a \wedge b \wedge c\right)$$
    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))
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