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Boolean calcul

Operator

Or (addition)

Noted a + b

a	b	R
0	0	0
0	1	1
1	0	1
1	1	1

And (multiplication)

Noted a.b or ab

a	b	R
0	0	0
0	1	0
1	0	0
1	1	1

complement (not)

Invert of a boolean specify with a dash above the variable: ā

a	ā
0	1
1	0

Properties

Neutral element

Element that does not influence the result.

or (addition): 0 because a + 0 = a
and (multiplication): 1 because a * 1 = a

Absorbing element

Element that make an operation equal to itself.

or (addition): 1 because a + 1 = 1
and (multiplication): 0 because a * 0 = 0

Absorbing with the complement

or (addition): a + ā = 1
and (multiplication): a.ā = 0

Indenpotence

or: a + a = a
and: a.a = a

Commutative law

a + b = b + a a.b = b.a

Associative law

(a + b) + c = a + (b + c) (ab)c = a(bc)

Distributive law

a(b + c) = ab + ac ab + ac = a(b + c)

Laws

Morgan law

first law
```
_____
a + b = ā.b̄
```
Second law
```
___
a.b = ā + b̄
```

Karnault

Faire le tableau de toutes les possibilités

groupement de 8, 4 ou 2 en verticale ou horizontale (toujours prendres le plus grand possible) On regarde les changement d’état d’une variable qui élimine la varibale.

Tableau 2 valeurs

ex: ab + ab̄ + āb | | b | b̄ | | a | x | x | | ā | x | |

Ligne 1 = a Colonne 1 = b resultat = a + b

Un groupement de 4 = 1 Un groupement de 8 = 1

Cas ab + a

On met deux croix dans la ligne a car a(b + b̄) = 1

	b	b̄
a	x	x
ā	x

Tableau 4 valeurs

marchent a cote et sur les extremités

autre

repose sur ce principe: ab + ab = ab donc: ab = ab + ab

ab + ab̄