\(S = (A ⊕ B) ⋅ (B ⊙ \overline{C})\)

1.Definicions de les operacions:

  • \(X ⊕ Y = (X⋅ \overline{Y}) + (\overline{X} ⋅ Y)\)
  • \(X ⊙ Y = (X ⋅ Y) + (\overline{X} ⋅ \overline{Y})\)

2. Substituint:

  • \(A ⊕ B = (A ⋅ \overline{B}) + (\overline{A} ⋅ B)\)
  • \(B ⊙ \overline{C} = (B ⋅ \overline{C}) + (\overline{B} ⋅ C)\)

3. Per tant,

\(S = [(A ⋅ \overline{B}) + (\overline{A} ⋅ B)] · [(B ⋅ \overline{C} ) + (\overline{B} ⋅ C)]\)

4. Propietat distributiva:

\(S = (A ⋅ \overline{B} ⋅ B ⋅ \overline{C}) + (A ⋅ \overline{B} ⋅ \overline{B} ⋅ C) + (\overline{A} ⋅ B ⋅ B ⋅ \overline{C}) + (\overline{A}⋅ B ⋅ \overline{B} ⋅ C)\)

5. Apliquem les regles de l'Àlgebra de Boole per simplificar cada terme:

  • \(X ⋅ \overline{X} = 0\)

  • \(X ⋅ X = X\)

  • \(X · 0 = 0\)

6. Així:

  • \((A ⋅ \overline{B} ⋅ B ⋅ \overline{C}) = 0\)

  • \((A ⋅ \overline{B} ⋅ \overline{B} ⋅ C) = (A ⋅ \overline{B} ⋅ C)\)

  • \((\overline{A} ⋅ B ⋅ B ⋅ \overline{C}) = (\overline{A} ⋅ B ⋅ \overline{C})\)

  • \((\overline{A} ⋅ B ⋅ \overline{B} ⋅ C) =0\)

7. Substituïm:

\(S = (A ⋅ \overline{B} ⋅ C) + (\overline{A} ⋅ B ⋅ \overline{C})\)

FORMA CANÒNICA DE SUMA DE PRODUCTES (S.O.P)

Darrera modificació: dijous, 1 d’octubre 2026, 13:49