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Lógica matemática/ (R→(G∧H))∧(¬R→(¬G∧¬H))

Expresión (R→(G∧H))∧(¬R→(¬G∧¬H))

El profesor se sorprenderá mucho al ver tu solución correcta😉

⌨

    Solución

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