Boolean Algebra Laws
Reference for Boolean algebra laws: De Morgan, identity, complement, and absorption.
Covers AND, OR, NOT, XOR, and NAND truth tables for digital logic.
The Formula
Identity: A + 0 = A, A · 1 = A
Null: A + 1 = 1, A · 0 = 0
Complement: A + A' = 1, A · A' = 0
De Morgan's: (A · B)' = A' + B', (A + B)' = A' · B'
Null: A + 1 = 1, A · 0 = 0
Complement: A + A' = 1, A · A' = 0
De Morgan's: (A · B)' = A' + B', (A + B)' = A' · B'
Boolean algebra is the mathematics of digital logic. It uses only two values (0 and 1) and three basic operations: AND (·), OR (+), and NOT (').
Variables
| Symbol | Meaning |
|---|---|
| A, B | Boolean variables (either 0 or 1) |
| + | OR operation |
| · | AND operation |
| ' | NOT operation (complement) |
| 0 | False / Low |
| 1 | True / High |
Example 1
Simplify: A · (A + B)
Apply distribution: A · A + A · B
A · A = A (idempotent law)
A + A · B = A (absorption law)
Result: A
Example 2
Apply De Morgan's law to: (A · B · C)'
(A · B · C)' = A' + B' + C'
The complement of an AND becomes an OR of the complements
When to Use It
Use Boolean algebra when:
- Designing and simplifying digital logic circuits
- Writing conditional logic in programming
- Optimizing database queries with AND/OR conditions
- Analyzing truth tables and logic gates
Key Notes
- Three fundamental operations: AND (·), OR (+), and NOT (¬). Every Boolean function — no matter how complex — can be built from these three operations.
- De Morgan's Laws: ¬(A · B) = ¬A + ¬B and ¬(A + B) = ¬A · ¬B. These are essential for simplifying logic circuits and converting between AND/OR forms. "Break the bar, change the operator."
- Idempotent and absorption laws: A · A = A, A + A = A (idempotent). A · (A + B) = A, A + (A · B) = A (absorption). These simplify redundant logic.
- Universal gates: Any Boolean function can be built using only NAND gates (or only NOR gates). This is why microchips are often designed entirely with one gate type.
- XOR is not a fundamental gate: XOR (exclusive OR, A ⊕ B) is a derived operation: A ⊕ B = (A · ¬B) + (¬A · B). XOR is true when exactly one input is true — widely used in encryption and parity checks.