Boolean algebra and logic gates
Boolean algebra
Boolean algebra is a mathematical structure that operates on true (1) and false (0) values, serving as the foundation for digital circuits and computer logic.
A and B can be equal 0 or 1.
A = not A, contradiction of A, A reversed
Statement 1. - Commutative Law
A + B = B + A
A * B = B * A
Statement 2. - Associative Law
(A + B) + C = A + (B + C)
Statement 3.
A * (B + C) = A * B + A * C
A + (B * C) = (A + B) * (A + C)
Statement 4. - Idempotent Law
A + A = A
A * A = A
Statement 5.
A * B + A * B = A
(A + B) * (A + B) = A
Statement 6. - Absorption Law
A + A * B = A
A * (A + B) = 1
Statement 7. - Identity Law
0 + A = A
0 * A = 0
Statement 8.
1 + A = 1
1 * A = A
Statement 9.
A + A = 1
A * A = 0
Statement 10.
A + A * B = A + B
A * (A + B) = A * B
Statement 11. - De Morgan’s Law
A + B = A * B
A * B = A + B
Logic gates
Logic gates are the fundamental building blocks of digital circuits that perform logical operations such as AND, OR, and NOT. They are used for processing binary signals.
Y is the result of letting A and B through a given logic gate.
| OR | AND | XOR | NOT | NOR | NAND | XNOR | ||||||||||||||
| Y = A + B | Y = A * B | Y = A * B + A * B | Y = A | Y = A + B | Y = A * B | Y = A * B + A * B | ||||||||||||||
| A | B | Y | A | B | Y | A | B | Y | A | B | A | B | Y | A | B | Y | A | B | Y | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | |
| 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | |
| 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | |||
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | |||
The NOR, NAND, and XNOR gates are the negations (NOT) of OR, AND, and XOR.
| A | B | A * B | A * B |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 |