Presentiamo adesso una carrellata delle principali equivalenze logiche. Per ciascuna equivalenza ho riportato l'espressione funzionale, quella booleana ed il relativo schema circuitale simbolico.

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Espressioni che coinvolgono \( 1\) solo operatore


Neutro somma

$$ x + 0 = x $$ $$ \mathrm{OR}(x, 0) = x $$


Annullamento

$$ x \cdot 0 = 0 $$ $$ \mathrm{AND}(x, 0) = 0 $$


Attivazione

$$ x + 1 = 1 $$ $$ \mathrm{OR}(x, 1) = 1 $$


Neutro Prodotto

$$ x \cdot 1 = x $$ $$ \mathrm{AND}(x, 1) = x $$


Idempotenza somma

$$ x + x = x $$ $$ \mathrm{OR}(x, x) = x $$


Idempotenza prodotto

$$ x \cdot x = x $$ $$ \mathrm{AND}(x, x) = x $$


Convoluzione

$$ x + \overline{x} = 1 $$ $$ \mathrm{OR}(x, \overline{x}) = 1 $$


Convoluzione

$$ x \cdot \overline{x} = 0 $$ $$ \mathrm{AND}(x, \overline{x}) = 0 $$


Espressioni che coinvolgono \( 2\) operatori

Associativa (sonna)


$$ x + (y + z) = (x + y) + z $$ $$ \mathrm{OR}\bigl(x, \mathrm{OR}(y, z)\bigr) = \mathrm{OR}\bigl(\mathrm{OR}(x, y), z\bigr) $$


$$ x \cdot y \cdot z $$ $$ \mathrm{AND}(x, y, z) $$

Associativa (prodotto)


$$ x \cdot (y \cdot z) = (x \cdot y) \cdot z $$ $$ \mathrm{AND}\bigl(x, \mathrm{AND}(y, z)\bigr) = \mathrm{AND}\bigl(\mathrm{AND}(x, y), z\bigr) $$


$$ x \cdot y \cdot z $$ $$ \mathrm{AND}(x, y, z) $$


Altre espressioni

Distributiva (prodotto-somma)


$$ x \cdot (y + z) $$ $$ \mathrm{AND}\bigl(x, \mathrm{OR}(y, z)\bigr) $$


$$ (x \cdot y) + (x \cdot z) $$ $$ \mathrm{AND}\bigl(x, y\bigr) + \mathrm{AND}\bigl(x, y\bigr) $$

Distributiva (somma-prodotto)


$$ x + (y \cdot z) $$ $$ \mathrm{OR}\bigl(x, \mathrm{AND}(y, z)\bigr) $$


$$ (x + y) \cdot (x + z) $$ $$ \mathrm{OR}\bigl(x, y\bigr) + \mathrm{OR}\bigl(x, y\bigr) $$


Negazione

$$ \overline{x} $$ $$ \mathrm{NOT}(x) $$


Buffer

$$ x $$


Complemento

$$ \overline{\overline{x}} = x $$ $$ \mathrm{NOT}\bigr(\mathrm{NOT}(x)\bigl) $$


Teoremi di De Morgan

$$ \overline{x+y} = \overline{x} \cdot \overline{y} $$ $$ \mathrm{NOT}\Bigl(\mathrm{OR}\bigl(x, y\bigr) \Bigr) = \mathrm{AND}\Bigl( \mathrm{NOT}\bigl(x\bigr), \mathrm{NOT}\bigl(y\bigr) \Bigl) $$




$$ \overline{x\cdot y} = \overline{x} + \overline{y} $$ $$ \mathrm{NOT}\Bigl(\mathrm{AND}\bigl(x, y\bigr) \Bigr) = \mathrm{OR}\Bigl( \mathrm{NOT}\bigl(x\bigr), \mathrm{NOT}\bigl(y\bigr) \Bigl) $$




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