Sr Examen
Expresión avb⇒((a⇒c)&(b⇒d)⇒cvd)
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Solución
Ha introducido
[src]
(a∨b)⇒(((a⇒c)∧(b⇒d))⇒(c∨d))
$$\left(a \vee b\right) \Rightarrow \left(\left(\left(a \Rightarrow c\right) \wedge \left(b \Rightarrow d\right)\right) \Rightarrow \left(c \vee d\right)\right)$$
Solución detallada
$$a \Rightarrow c = c \vee \neg a$$
$$b \Rightarrow d = d \vee \neg b$$
$$\left(a \Rightarrow c\right) \wedge \left(b \Rightarrow d\right) = \left(c \wedge d\right) \vee \left(c \wedge \neg b\right) \vee \left(d \wedge \neg a\right) \vee \left(\neg a \wedge \neg b\right)$$
$$\left(\left(a \Rightarrow c\right) \wedge \left(b \Rightarrow d\right)\right) \Rightarrow \left(c \vee d\right) = a \vee b \vee c \vee d$$
$$\left(a \vee b\right) \Rightarrow \left(\left(\left(a \Rightarrow c\right) \wedge \left(b \Rightarrow d\right)\right) \Rightarrow \left(c \vee d\right)\right) = 1$$
$$b \Rightarrow d = d \vee \neg b$$
$$\left(a \Rightarrow c\right) \wedge \left(b \Rightarrow d\right) = \left(c \wedge d\right) \vee \left(c \wedge \neg b\right) \vee \left(d \wedge \neg a\right) \vee \left(\neg a \wedge \neg b\right)$$
$$\left(\left(a \Rightarrow c\right) \wedge \left(b \Rightarrow d\right)\right) \Rightarrow \left(c \vee d\right) = a \vee b \vee c \vee d$$
$$\left(a \vee b\right) \Rightarrow \left(\left(\left(a \Rightarrow c\right) \wedge \left(b \Rightarrow d\right)\right) \Rightarrow \left(c \vee d\right)\right) = 1$$
Tabla de verdad
+---+---+---+---+--------+ | a | b | c | d | 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 | 1 | +---+---+---+---+--------+ | 0 | 1 | 1 | 1 | 1 | +---+---+---+---+--------+ | 1 | 0 | 0 | 0 | 1 | +---+---+---+---+--------+ | 1 | 0 | 0 | 1 | 1 | +---+---+---+---+--------+ | 1 | 0 | 1 | 0 | 1 | +---+---+---+---+--------+ | 1 | 0 | 1 | 1 | 1 | +---+---+---+---+--------+ | 1 | 1 | 0 | 0 | 1 | +---+---+---+---+--------+ | 1 | 1 | 0 | 1 | 1 | +---+---+---+---+--------+ | 1 | 1 | 1 | 0 | 1 | +---+---+---+---+--------+ | 1 | 1 | 1 | 1 | 1 | +---+---+---+---+--------+