Boolean Algebra: Theorems & Properties
The algebra rules that let you shrink a logic expression without changing what it does.
Description
A set of algebraic identities over {0,1} with operators AND (·), OR (+), and NOT (′). Simpler expressions mean fewer gates: less area, less delay, less power. Apply theorems repeatedly to factor, absorb, or cancel terms until no rule applies.
- Each law has a dual: swap AND↔OR and 0↔1 to get the partner law.
- Single-variable laws collapse redundancy; two/three-variable laws restructure terms.
- DeMorgan converts between AND/OR forms — the basis of NAND/NOR design.
- Absorption: A + AB = A (a term is swallowed by a simpler one).
- Combining: AB + AB′ = A (a variable that appears both true and complemented drops out).
- DeMorgan: (AB)′ = A′ + B′ and (A + B)′ = A′B′.
- What: A set of algebraic identities over {0,1} with operators AND (·), OR (+), and NOT (′).
- Why: Simpler expressions mean fewer gates: less area, less delay, less power.
- How: Apply theorems repeatedly to factor, absorb, or cancel terms until no rule applies.
- Where: Pre-K-map simplification, deriving canonical forms, and HDL expression cleanup.
At a glance
What
A set of algebraic identities over {0,1} with operators AND (·), OR (+), and NOT (′).
Why
Simpler expressions mean fewer gates: less area, less delay, less power.
How
Apply theorems repeatedly to factor, absorb, or cancel terms until no rule applies.
Where
Pre-K-map simplification, deriving canonical forms, and HDL expression cleanup.
When
Before committing a logic equation to a circuit or RTL — minimize first.
Think of it like…
Like cancelling terms in algebra to shorten an expression — absorption (A+AB=A) is the logic version of noticing a clause is already implied, so you cross it out.
How to read the laws
- Each law has a dual: swap AND↔OR and 0↔1 to get the partner law.
- Single-variable laws collapse redundancy; two/three-variable laws restructure terms.
- DeMorgan converts between AND/OR forms — the basis of NAND/NOR design.
Most-used in practice
- Absorption: A + AB = A (a term is swallowed by a simpler one).
- Combining: AB + AB′ = A (a variable that appears both true and complemented drops out).
- DeMorgan: (AB)′ = A′ + B′ and (A + B)′ = A′B′.
Boolean theorems (with duals)
| Law | OR form | AND form (dual) |
|---|---|---|
| Identity | A + 0 = A | A · 1 = A |
| Null / Dominance | A + 1 = 1 | A · 0 = 0 |
| Idempotent | A + A = A | A · A = A |
| Complement | A + A′ = 1 | A · A′ = 0 |
| Involution | (A′)′ = A | (A′)′ = A |
| Commutative | A + B = B + A | A · B = B · A |
| Associative | A+(B+C)=(A+B)+C | A(BC)=(AB)C |
| Distributive | A+BC=(A+B)(A+C) | A(B+C)=AB+AC |
| Absorption | A + AB = A | A(A+B) = A |
| DeMorgan | (A+B)′ = A′B′ | (AB)′ = A′+B′ |
Every row's two forms are duals: swap + ↔ · and 0 ↔ 1.
Worked simplification
| Step | Expression | Law used |
|---|---|---|
| start | A + A′B | — |
| 1 | (A + A′)(A + B) | distributive |
| 2 | 1 · (A + B) | complement |
| 3 | A + B | identity |
A + A′B = A + B — three theorems, no truth table needed.
Verify a law on real gates
▶ live simulatorClick a terminal (A/B) to toggle it · glowing wires carry a logic 1 · the lamp is output Y
| A | B | Y |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
The 5 Whys
- 1
Why simplify Boolean expressions? To use fewer logic gates.
- 2
Why fewer gates? Less silicon area and lower cost per chip.
- 3
Why does delay drop too? Shorter gate chains mean signals settle faster.
- 4
Why does power drop? Fewer switching nodes dissipate less dynamic energy.
- 5
Root cause: the theorems remove logical redundancy the truth table can't see directly.
Cheat sheet
Working principle
- Apply theorems repeatedly to factor, absorb, or cancel terms until no rule applies.
- A set of algebraic identities over {0,1} with operators AND (·), OR (+), and NOT (′).
Formulas & Boolean expressions
- Absorption: A + AB = A (a term is swallowed by a simpler one).
- Combining: AB + AB′ = A (a variable that appears both true and complemented drops out).
- DeMorgan: (AB)′ = A′ + B′ and (A + B)′ = A′B′.
Key facts
- Each law has a dual: swap AND↔OR and 0↔1 to get the partner law.
- Absorption: A + AB = A (a term is swallowed by a simpler one).
Why it exists
- Root cause: the theorems remove logical redundancy the truth table can't see directly.