These are the identities every Boolean simplification — by hand, by K-map, or by an algorithm like Quine-McCluskey — ultimately relies on. Each one below has been checked by exhaustive truth table, not just stated from memory.
Identity & null laws
| Law | Rule |
|---|---|
| Identity | A + 0 = A A · 1 = A |
| Null (domination) | A + 1 = 1 A · 0 = 0 |
| Idempotent | A + A = A A · A = A |
| Complement | A + A′ = 1 A · A′ = 0 |
| Double negation | (A′)′ = A |
Commutative, associative, distributive
| Law | Rule |
|---|---|
| Commutative | A + B = B + A A·B = B·A |
| Associative | (A+B)+C = A+(B+C) (A·B)·C = A·(B·C) |
| Distributive | A·(B+C) = A·B + A·C |
| Distributive (OR over AND) | A+(B·C) = (A+B)·(A+C) |
That second distributive law looks unfamiliar coming from ordinary arithmetic, where it has no equivalent — but it holds in Boolean algebra specifically, and it's exactly the identity behind converting between SOP and POS forms.
Absorption
| Law | Rule |
|---|---|
| Absorption | A + A·B = A A·(A+B) = A |
| Absorption variant | A + A′·B = A + B |
The absorption variant is the one that actually does work during minimization — it's the algebraic reason a larger group with fewer literals is always at least as good as keeping a smaller redundant term around.
De Morgan's theorem
| Law | Rule |
|---|---|
| De Morgan's, first form | (A + B)′ = A′ · B′ |
| De Morgan's, second form | (A · B)′ = A′ + B′ |
This is the exact mechanism behind converting a grouped set of 0s into a proper POS sum term — complement each literal and flip the implicit AND to an OR. Full walkthrough in SOP vs. POS Explained.
Consensus theorem
| Law | Rule |
|---|---|
| Consensus | A·B + A′·C + B·C = A·B + A′·C |
The B·C term is always redundant whenever A·B and A′·C are both present — it never changes the function's output, only its length. This is the algebraic version of what a K-map grouping catches automatically: a term entirely covered by the combination of two others contributes nothing and can be dropped.