Hukum Aljabar Boolean
1. Hukum Komutatif
(a) A + B = B + A
Tabel Kebenaran:
| A | B | A + B | B + A |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 |
(b) A B = B A
Tabel Kebenaran:
| A | B | AB | BA |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
2. Hukum Asosiatif
(a) (A + B) + C = A + (B + C)
Tabel Kebenaran:
| A | B | C | A + B | B + C | (A+B)+C | A+(B+C) |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 | 0 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
(b) (A B) C = A (B C)
Tabel Kebenaran:
| A | B | C | AB | BC | (AB)C | A(BC) |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
3. Hukum Distributif
(a) A (B + C) = A B + A C
Tabel Kebenaran:
| A | B | C | B +C | AB | AC | A(B+C) | (AB)+(AC) |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
(b) A + (B C) = (A + B) (A + C)
Tabel Kebenaran:
| A | B | C | BC | A+B | A+C | A+(BC) | (A+B)(A+C) |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
4. Hukum Identity
(a) A + A = A
Tabel Kebenaran:
| A | A + A |
| 0 | 0 |
| 0 | 0 |
| 1 | 1 |
| 1 | 1 |
(b) A A = A
Tabel Kebenaran:
| A | A A |
| 0 | 0 |
| 0 | 0 |
| 1 | 1 |
| 1 | 1 |
5.
(a) AB + A B’
Tabel Kebenaran:
| A | B | B' | A B | A B' | AB+AB' |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 | 0 | 1 |
(b) (A+B)(A+B’)
Tabel Kebenaran:
| A | B | B' | A+B | A+B' |
| 0 | 0 | 1 | 0 | 1 |
| 0 | 1 | 0 | 1 | 0 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 |
6. Hukum Redudansi
(a) A + A B = A
Tabel Kebenaran:
| A | B | A B | A + A B |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 |
(b) A (A + B) = A
Tabel Kebenaran:
| A | B | A + B | A (A + B) |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 |
7
(a) 0 + A = A
Tabel Kebenaran:
| A | 0 + A |
| 0 | 0 |
| 0 | 0 |
| 1 | 1 |
| 1 | 1 |
(b) 0 A = 0
Tabel Kebenaran:
| A | 0 A | 0 |
| 0 | 0 | 0 |
| 0 | 0 | 0 |
| 1 | 0 | 0 |
| 1 | 0 | 0 |
8
(a) 1 + A = 1
Tabel Kebenaran:
| A | 1 + A | 1 |
| 0 | 1 | 1 |
| 0 | 1 | 1 |
| 1 | 1 | 1 |
| 1 | 1 | 1 |
(b) 1 A = A
Tabel Kebenaran:
| A | 1 A |
| 0 | 0 |
| 0 | 0 |
| 1 | 1 |
| 1 | 1 |
9
(a) A’ + A = 1
Tabel Kebenaran:
| A | A' | A' | 1 |
| 0 | 1 | 1 | 1 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 |
| 1 | 0 | 1 | 1 |
(b) A’ A=0
Tabel Kebenaran:
| A | A' | A'A | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
10
(a) A + A’ B =A + B
Tabel Kebenaran:
| A | B | A' | A' B | A+B | A+A' B |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 1 | 1 |
(b) A (A’ + B) = AB
Tabel Kebenaran:
| A | B | A' | A'+B | A B | A(A'+B) |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 | 1 |
11. TheoremaDe Morgan's
(a) (A’+B’)= A’B’
Tabel Kebenaran:
| A | B | A' | B' | A+B | (A+B)' | A' B' |
| 0 | 0 | 1 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 | 1 | 0 | 0 |
(b) (A’B’) = A’ + B’
Tabel Kebenaran:
| A | B | A' | B' | A B | (AB)' | A'+B' |
| 0 | 0 | 1 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 | 1 | 0 | 0 |
Quiz Aljabar Boolean
1. Give the relationship that represents the dual of the Boolean property A + 1 = 1?
(Note: * = AND, + = OR and ' = NOT)
1. A * 1 = 1
2. A * 0 = 0
3. A + 0 = 0
4. A * A = A
5. A * 1 = 1
2. Give the best definition of a literal?
1. A Boolean variable
2. The complement of a Boolean variable
3. 1 or 2
4. A Boolean variable interpreted literally
5. The actual understanding of a Boolean variable
3. Simplify the Boolean expression (A+B+C)(D+E)' + (A+B+C)(D+E) and choose the best answer.
1. A + B + C
2. D + E
3. A'B'C'
4. D'E'
5. None of the above
4. Which of the following relationships represents the dual of the Boolean property x + x'y = x + y?
1. x'(x + y') = x'y'
2. x(x'y) = xy
3. x*x' + y = xy
4. x'(xy') = x'y'
5. x(x' + y) = xy
5. Given the function F(X,Y,Z) = XZ + Z(X'+ XY), the equivalent most simplified Boolean representation for F is:
1. Z + YZ
2. Z + XYZ
3. XZ
4. X + YZ
5. None of the above
6. Which of the following Boolean functions is algebraically complete?
1. F = xy
2. F = x + y
3. F = x'
4. F = xy + yz
5. F = x + y'
7. Simplification of the Boolean expression (A + B)'(C + D + E)' + (A + B)' yields which of the following results?
1. A + B
2. A'B'
3. C + D + E
4. C'D'E'
5. A'B'C'D'E'
8. Given that F = A'B'+ C'+ D'+ E', which of the following represent the only correct expression for F'?
1. F'= A+B+C+D+E
2. F'= ABCDE
3. F'= AB(C+D+E)
4. F'= AB+C'+D'+E'
5. F'= (A+B)CDE
9. An equivalent representation for the Boolean expression A' + 1 is
1. A
2. A'
3. 1
4. 0
10. Simplification of the Boolean expression AB + ABC + ABCD + ABCDE + ABCDEF yields which of the following results?
1. ABCDEF
2. AB
3. AB + CD + EF
4. A + B + C + D + E + F
5. A + B(C+D(E+F))
