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Lógica matemática/ ¬(A→(B∧C)→D)

Expresión ¬(A→(B∧C)→D)

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    Solución

    Ha introducido [src] 
    ¬((a⇒(b∧c))⇒d)
    $$\left(a \Rightarrow \left(b \wedge c\right)\right) \not\Rightarrow d$$
    Solución detallada
    $$a \Rightarrow \left(b \wedge c\right) = \left(b \wedge c\right) \vee \neg a$$
    $$\left(a \Rightarrow \left(b \wedge c\right)\right) \Rightarrow d = d \vee \left(a \wedge \neg b\right) \vee \left(a \wedge \neg c\right)$$
    $$\left(a \Rightarrow \left(b \wedge c\right)\right) \not\Rightarrow d = \neg d \wedge \left(b \vee \neg a\right) \wedge \left(c \vee \neg a\right)$$
    Simplificación [src]
    $$\neg d \wedge \left(b \vee \neg a\right) \wedge \left(c \vee \neg a\right)$$ 
    (¬d)∧(b∨(¬a))∧(c∨(¬a))
    Tabla de verdad 
    +---+---+---+---+--------+
| a | b | c | d | result |
+===+===+===+===+========+
| 0 | 0 | 0 | 0 | 1      |
+---+---+---+---+--------+
| 0 | 0 | 0 | 1 | 0      |
+---+---+---+---+--------+
| 0 | 0 | 1 | 0 | 1      |
+---+---+---+---+--------+
| 0 | 0 | 1 | 1 | 0      |
+---+---+---+---+--------+
| 0 | 1 | 0 | 0 | 1      |
+---+---+---+---+--------+
| 0 | 1 | 0 | 1 | 0      |
+---+---+---+---+--------+
| 0 | 1 | 1 | 0 | 1      |
+---+---+---+---+--------+
| 0 | 1 | 1 | 1 | 0      |
+---+---+---+---+--------+
| 1 | 0 | 0 | 0 | 0      |
+---+---+---+---+--------+
| 1 | 0 | 0 | 1 | 0      |
+---+---+---+---+--------+
| 1 | 0 | 1 | 0 | 0      |
+---+---+---+---+--------+
| 1 | 0 | 1 | 1 | 0      |
+---+---+---+---+--------+
| 1 | 1 | 0 | 0 | 0      |
+---+---+---+---+--------+
| 1 | 1 | 0 | 1 | 0      |
+---+---+---+---+--------+
| 1 | 1 | 1 | 0 | 1      |
+---+---+---+---+--------+
| 1 | 1 | 1 | 1 | 0      |
+---+---+---+---+--------+
    FNC [src]
    Ya está reducido a FNC
    $$\neg d \wedge \left(b \vee \neg a\right) \wedge \left(c \vee \neg a\right)$$ 
    (¬d)∧(b∨(¬a))∧(c∨(¬a))
    FNDP [src]
    $$\left(\neg a \wedge \neg d\right) \vee \left(b \wedge c \wedge \neg d\right)$$ 
    ((¬a)∧(¬d))∨(b∧c∧(¬d))
    FND [src]
    $$\left(\neg a \wedge \neg d\right) \vee \left(b \wedge c \wedge \neg d\right) \vee \left(b \wedge \neg a \wedge \neg d\right) \vee \left(c \wedge \neg a \wedge \neg d\right)$$ 
    ((¬a)∧(¬d))∨(b∧c∧(¬d))∨(b∧(¬a)∧(¬d))∨(c∧(¬a)∧(¬d))
    FNCD [src]
    $$\neg d \wedge \left(b \vee \neg a\right) \wedge \left(c \vee \neg a\right)$$ 
    (¬d)∧(b∨(¬a))∧(c∨(¬a))
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