Boolean Algebra#
Boolean algebra is the algebra of two-valued logic: variables that are only ever 0 or 1, combined with three operations — AND (·), OR (+), and NOT (′). It is the lever between a truth table and a circuit, because it lets a logic expression be manipulated and simplified on paper before a single gate is committed. Fewer terms mean fewer gates, fewer levels of logic, less delay, and less power.
The Identities#
The manipulation rests on a small set of laws. Most come in dual pairs — swap AND for OR and 0 for 1 and one identity becomes the other:
| Law | AND form | OR form |
|---|---|---|
| Identity | A · 1 = A | A + 0 = A |
| Null | A · 0 = 0 | A + 1 = 1 |
| Idempotent | A · A = A | A + A = A |
| Complement | A · A′ = 0 | A + A′ = 1 |
| Absorption | A · (A + B) = A | A + A·B = A |
| Distributive | A·(B + C) = A·B + A·C | A + B·C = (A + B)·(A + C) |
Add the obvious ones — commutative and associative ordering, and involution ((A′)′ = A) — and these are enough to transform any expression into any equivalent one. The second distributive law (A + BC = (A+B)(A+C)) is the one that surprises people, because ordinary arithmetic has no equivalent.
Where It Fits#
Algebraic simplification is the general-purpose tool: it works for any number of variables and it is what a synthesis tool does internally. By hand it is powerful but easy to get lost in, which is why two special cases get their own treatment. De Morgan’s theorems handle inversion across a whole gate and are the single most-used identities in practice, and Karnaugh maps turn simplification of small functions from algebra into pattern-spotting. All three are doing the same job — trading a complicated expression for a simpler circuit that computes the identical truth table.