Booleova algebra

Základní operace

Logický součin (AND):

 A | B | A ⋅ B
---|---|-------
 0 | 0 |   0
 0 | 1 |   0
 1 | 0 |   0
 1 | 1 |   1

Logický součet (OR):

 A | B | A + B 
---|---|-------
 0 | 0 |   0
 0 | 1 |   1
 1 | 0 |   1
 1 | 1 |   1

Negace (NOT):

 A | ¬A
---|----
 0 |  1
 1 |  0

Axiomy Booleovy algebry

Booleova algebra se řídí několika základními axiomy:

• Komutativita:
  • A ⋅ B = B ⋅ A
  • A + B = B + A
• Asociativita:
  • (A ⋅ B) ⋅ C = A ⋅ (B ⋅ C)
  • (A + B) + C = A + (B + C)
• Distributivita:
  • A ⋅ (B + C) = (A ⋅ B) + (A ⋅ C)
  • A + (B ⋅ C) = (A + B) ⋅ (A + C)
• Existence neutrálních prvků (identita):
  • A ⋅ 1 = A
  • A + 0 = A
• Existence inverzních prvků (doplněk):
  • A ⋅ ¬A = 0
  • A + ¬A = 1

Teorémy Booleovy algebry

Některá odvozená pravidla:

• Idempotence:
  • A + A = A
  • A ⋅ A = A
• Dominance (nulový prvek):
  • A + 1 = 1
  • A ⋅ 0 = 0
• Dvojitá negace:
  • ¬(¬A) = A
• De Morganovy zákony:
  • ¬(A + B) = ¬A ⋅ ¬B
  • ¬(A ⋅ B) = ¬A + ¬B
• Absorpční zákony:
  • A + (A ⋅ B) = A
  • A ⋅ (A + B) = A
• Absorpce negace:
  • A + (A' ⋅ B) = A + B
  • A ⋅ (A' + B) = A ⋅ B