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Lógica matemática/ ¬(((A∨B)⇒C∧D)∧((D∨B)⇒F))

Expresión ¬(((A∨B)⇒C∧D)∧((D∨B)⇒F))

El profesor se sorprenderá mucho al ver tu solución correcta😉

⌨

    Solución

    Ha introducido [src] 
    ¬(((b∨d)⇒f)∧((a∨b)⇒(c∧d)))
    $$\neg \left(\left(\left(a \vee b\right) \Rightarrow \left(c \wedge d\right)\right) \wedge \left(\left(b \vee d\right) \Rightarrow f\right)\right)$$
    Solución detallada
    $$\left(b \vee d\right) \Rightarrow f = f \vee \left(\neg b \wedge \neg d\right)$$
    $$\left(a \vee b\right) \Rightarrow \left(c \wedge d\right) = \left(c \vee \neg a\right) \wedge \left(c \vee \neg b\right) \wedge \left(d \vee \neg a\right) \wedge \left(d \vee \neg b\right)$$
    $$\left(\left(a \vee b\right) \Rightarrow \left(c \wedge d\right)\right) \wedge \left(\left(b \vee d\right) \Rightarrow f\right) = \left(c \vee \neg a\right) \wedge \left(c \vee \neg b\right) \wedge \left(d \vee \neg a\right) \wedge \left(d \vee \neg b\right) \wedge \left(f \vee \neg d\right)$$
    $$\neg \left(\left(\left(a \vee b\right) \Rightarrow \left(c \wedge d\right)\right) \wedge \left(\left(b \vee d\right) \Rightarrow f\right)\right) = \left(a \wedge \neg c\right) \vee \left(a \wedge \neg d\right) \vee \left(b \wedge \neg c\right) \vee \left(b \wedge \neg d\right) \vee \left(d \wedge \neg f\right)$$
    Simplificación [src]
    $$\left(a \wedge \neg c\right) \vee \left(a \wedge \neg d\right) \vee \left(b \wedge \neg c\right) \vee \left(b \wedge \neg d\right) \vee \left(d \wedge \neg f\right)$$ 
    (a∧(¬c))∨(a∧(¬d))∨(b∧(¬c))∨(b∧(¬d))∨(d∧(¬f))
    Tabla de verdad 
    +---+---+---+---+---+--------+
| a | b | c | d | f | result |
+===+===+===+===+===+========+
| 0 | 0 | 0 | 0 | 0 | 0      |
+---+---+---+---+---+--------+
| 0 | 0 | 0 | 0 | 1 | 0      |
+---+---+---+---+---+--------+
| 0 | 0 | 0 | 1 | 0 | 1      |
+---+---+---+---+---+--------+
| 0 | 0 | 0 | 1 | 1 | 0      |
+---+---+---+---+---+--------+
| 0 | 0 | 1 | 0 | 0 | 0      |
+---+---+---+---+---+--------+
| 0 | 0 | 1 | 0 | 1 | 0      |
+---+---+---+---+---+--------+
| 0 | 0 | 1 | 1 | 0 | 1      |
+---+---+---+---+---+--------+
| 0 | 0 | 1 | 1 | 1 | 0      |
+---+---+---+---+---+--------+
| 0 | 1 | 0 | 0 | 0 | 1      |
+---+---+---+---+---+--------+
| 0 | 1 | 0 | 0 | 1 | 1      |
+---+---+---+---+---+--------+
| 0 | 1 | 0 | 1 | 0 | 1      |
+---+---+---+---+---+--------+
| 0 | 1 | 0 | 1 | 1 | 1      |
+---+---+---+---+---+--------+
| 0 | 1 | 1 | 0 | 0 | 1      |
+---+---+---+---+---+--------+
| 0 | 1 | 1 | 0 | 1 | 1      |
+---+---+---+---+---+--------+
| 0 | 1 | 1 | 1 | 0 | 1      |
+---+---+---+---+---+--------+
| 0 | 1 | 1 | 1 | 1 | 0      |
+---+---+---+---+---+--------+
| 1 | 0 | 0 | 0 | 0 | 1      |
+---+---+---+---+---+--------+
| 1 | 0 | 0 | 0 | 1 | 1      |
+---+---+---+---+---+--------+
| 1 | 0 | 0 | 1 | 0 | 1      |
+---+---+---+---+---+--------+
| 1 | 0 | 0 | 1 | 1 | 1      |
+---+---+---+---+---+--------+
| 1 | 0 | 1 | 0 | 0 | 1      |
+---+---+---+---+---+--------+
| 1 | 0 | 1 | 0 | 1 | 1      |
+---+---+---+---+---+--------+
| 1 | 0 | 1 | 1 | 0 | 1      |
+---+---+---+---+---+--------+
| 1 | 0 | 1 | 1 | 1 | 0      |
+---+---+---+---+---+--------+
| 1 | 1 | 0 | 0 | 0 | 1      |
+---+---+---+---+---+--------+
| 1 | 1 | 0 | 0 | 1 | 1      |
+---+---+---+---+---+--------+
| 1 | 1 | 0 | 1 | 0 | 1      |
+---+---+---+---+---+--------+
| 1 | 1 | 0 | 1 | 1 | 1      |
+---+---+---+---+---+--------+
| 1 | 1 | 1 | 0 | 0 | 1      |
+---+---+---+---+---+--------+
| 1 | 1 | 1 | 0 | 1 | 1      |
+---+---+---+---+---+--------+
| 1 | 1 | 1 | 1 | 0 | 1      |
+---+---+---+---+---+--------+
| 1 | 1 | 1 | 1 | 1 | 0      |
+---+---+---+---+---+--------+
    FND [src]
    Ya está reducido a FND
    $$\left(a \wedge \neg c\right) \vee \left(a \wedge \neg d\right) \vee \left(b \wedge \neg c\right) \vee \left(b \wedge \neg d\right) \vee \left(d \wedge \neg f\right)$$ 
    (a∧(¬c))∨(a∧(¬d))∨(b∧(¬c))∨(b∧(¬d))∨(d∧(¬f))
    FNCD [src]
    $$\left(a \vee b \vee d\right) \wedge \left(a \vee b \vee \neg f\right) \wedge \left(\neg c \vee \neg d \vee \neg f\right)$$ 
    (a∨b∨d)∧(a∨b∨(¬f))∧((¬c)∨(¬d)∨(¬f))
    FNC [src]
    $$\left(a \vee b \vee d\right) \wedge \left(a \vee b \vee \neg f\right) \wedge \left(d \vee \neg c \vee \neg d\right) \wedge \left(\neg c \vee \neg d \vee \neg f\right) \wedge \left(a \vee b \vee d \vee \neg c\right) \wedge \left(a \vee b \vee d \vee \neg d\right) \wedge \left(a \vee b \vee \neg c \vee \neg f\right) \wedge \left(a \vee b \vee \neg d \vee \neg f\right) \wedge \left(a \vee d \vee \neg c \vee \neg d\right) \wedge \left(a \vee \neg c \vee \neg d \vee \neg f\right) \wedge \left(b \vee d \vee \neg c \vee \neg d\right) \wedge \left(b \vee \neg c \vee \neg d \vee \neg f\right) \wedge \left(a \vee b \vee d \vee \neg c \vee \neg d\right) \wedge \left(a \vee b \vee \neg c \vee \neg d \vee \neg f\right)$$ 
    (a∨b∨d)∧(a∨b∨(¬f))∧(d∨(¬c)∨(¬d))∧(a∨b∨d∨(¬c))∧(a∨b∨d∨(¬d))∧((¬c)∨(¬d)∨(¬f))∧(a∨b∨(¬c)∨(¬f))∧(a∨b∨(¬d)∨(¬f))∧(a∨d∨(¬c)∨(¬d))∧(b∨d∨(¬c)∨(¬d))∧(a∨(¬c)∨(¬d)∨(¬f))∧(b∨(¬c)∨(¬d)∨(¬f))∧(a∨b∨d∨(¬c)∨(¬d))∧(a∨b∨(¬c)∨(¬d)∨(¬f))
    FNDP [src]
    $$\left(a \wedge \neg c\right) \vee \left(a \wedge \neg d\right) \vee \left(b \wedge \neg c\right) \vee \left(b \wedge \neg d\right) \vee \left(d \wedge \neg f\right)$$ 
    (a∧(¬c))∨(a∧(¬d))∨(b∧(¬c))∨(b∧(¬d))∨(d∧(¬f))
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