Boolean Algebra Guide
Truth Tables
| A | B | AND | OR | NAND | NOR | XOR | XNOR | NOT A |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 |
Boolean Laws
Identity Laws
A + 0 = A
A · 1 = A
Null Laws
A + 1 = 1
A · 0 = 0
Idempotent Laws
A + A = A
A · A = A
Complement Laws
A + A' = 1
A · A' = 0
Double Negation
(A')' = A
Commutative Laws
A + B = B + A
A · B = B · A
Associative Laws
A+(B+C) = (A+B)+C
A·(B·C) = (A·B)·C
Distributive Laws
A·(B+C) = A·B + A·C
A+(B·C) = (A+B)·(A+C)
De Morgan's Laws
(A+B)' = A'·B'
(A·B)' = A'+B'
Absorption Laws
A + A·B = A
A · (A+B) = A
Logic Gate Symbols
| Gate | Expression | Description |
|---|---|---|
| AND | Y = A · B | Output 1 only when all inputs are 1 |
| OR | Y = A + B | Output 1 when any input is 1 |
| NOT | Y = A' | Output is inverse of input |
| NAND | Y = (A·B)' | Inverse of AND |
| NOR | Y = (A+B)' | Inverse of OR |
| XOR | Y = A⊕B | Output 1 when inputs differ |
| XNOR | Y = (A⊕B)' | Output 1 when inputs are equal |