Teorema Aljabar Boolean

Berikut adalah beberapa teorema aljabar boolean yang biasa digunakan untuk menyederhanakan Rangkaian Logika .

1. Komutatif
   1. A + B = B+A
   2. A . B = B . A
2. Asosiatif
   1. ( A + B ) + C = A + ( B+ C )
   2. ( A . B ) . C = A . ( B . C )
3. Distributif
   1. A . ( B + C ) = ( A . B ) + ( A + C )
   2. A + ( B . C ) = ( A + B ) . ( A + C )
4. Identitas
   1. A + A = A
   2. A . A = A
5. Negasi
   1. ( A’ ) = A’
   2. ( A” ) = A
6. De Morgans
   1. ( A + B )’ = A’ . B’
   2. ( A . B )’ = A’ + B’
7. Tambahan
   1. 0 + A = A
   2. 1 . A = A
   3. 1 + A = 1
   4. 0 . A = 0
   5. A’ + A = 1
   6. A’ . A = 0
   7. A + A’ . B = A + B
   8. A . ( A’ + B ) = A . B

Contoh soal :

1. Sederhanakan rangkaian beriku :
   1. F = ( A + B’ + C’ ) ( A’ + B’C )
   2. F = A’B + ( CD)’ + AC’ + ACD

Jawab :

1. F = ( A + B’ + C’ ) ( A’ + B’C)

= AA’ + A’B’ + A’C’ + AB’C +B’B'C + B’CC’

= A’B’ + A’C’ + B’C ( 1 + A )

= A’B’ + A’C’ + B’C  (√)

2. F = A’B + ( CD)’ + AC’ + ACD

= A’B + C’ + D’ + AC’ + ACD

= A’B + ACD + D’ + C’ ( 1 + A )

= A’B + ACD + C’ + D’

= A’B + ACD + (CD)’ (√)