Boolean identities and duality

Q: What are the basic Boolean algebra identities?

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), involution (NOT (NOT A) = A), and absorption (A + A·B = A, A·(A + B) = A). They hold for any inputs and are the tools for simplifying by hand.

Q: What is the absorption law?

A + A·B = A (and its dual A·(A + B) = A). A term absorbs any product that already contains it, because whenever A·B is true A is already true, so the extra term cannot change the result. It is one of the most useful simplification levers.

The laws that simplify an expression by hand

4 min read

Boolean identities are the fixed laws (identity, null, idempotent, complement, and absorption) that hold for any inputs, and the duality principle says swapping every AND with OR and every 0 with 1 in a true identity yields another true identity, so together they let you simplify a logic expression by hand.

Builds onBoolean algebra

Boolean algebra introduced the operations and De Morgan's laws. This page collects the everyday identities you reach for when simplifying, and the duality shortcut that doubles every one of them, so you can carve an expression down before automating the hunt with a Karnaugh map.

The laws, grouped

Read · as AND and + as OR. Each law looks obvious on its own; the power is in chaining them. They come in pairs, one for AND and one for OR:

Identity: A·1 = A and A+0 = A. Combining with the neutral value changes nothing.
Null (annihilator): A·0 = 0 and A+1 = 1. The dominant value swallows everything.
Idempotent: A·A = A and A+A = A. Repeating a term is redundant.
Complement: A·A' = 0 and A+A' = 1. A thing and its opposite are never both true, and always cover all cases.
Involution: NOT (NOT A) = A. Two NOTs cancel (this is why AND is a NAND then a NOT).
Absorption: A + A·B = A and A·(A + B) = A. A term absorbs any product or sum that already contains it.

The duality principle

Notice the pairs. That is no accident: it is the duality principle. Take any true Boolean identity, swap every · with + and every 0 with 1, leave the variables and their NOTs alone, and the result is also true. One proof buys two laws.

A·1 = A has the dual A+0 = A.
A·0 = 0 has the dual A+1 = 1.
A + A·B = A (absorption) has the dual A·(A + B) = A.

Duality is not De Morgan. Duality only swaps · with + and 0 with 1; it never complements a variable, and its result is a *different* identity, not an equal expression. De Morgan also complements every variable and gives a real equality you can substitute. Use duality to remember twice as many laws; use De Morgan to push a NOT through a gate.

Worked example: absorption in action

Simplify A + A·B. The absorption law does it in one step, but it is worth seeing *why*. Factor A out: A + A·B = A·(1 + B). By the null law 1 + B = 1, so A·(1 + B) = A·1 = A. The A·B term added nothing: whenever A·B is true, A is already true, so the OR was never going to change.

A + A·B = A·(1 + B) = A·1 = A

That collapse of a two-gate expression to a single wire is a full half adder's worth of savings when it happens inside a real design. Chaining identities like this is exactly the simplification a Karnaugh map automates.

Try it

Simplify A·B + A·B' using the complement and identity laws. What single term remains?

Frequently asked

What are the basic Boolean algebra identities?

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), involution (NOT (NOT A) = A), and absorption (A + A·B = A, A·(A + B) = A). They hold for any inputs and are the tools for simplifying by hand.

What is the duality principle in Boolean algebra?

It says that if you take any true Boolean identity and swap every AND (·) with OR (+) and every 0 with 1, leaving the variables alone, the result is also true. That is why the laws come in AND/OR pairs: A·1 = A has the dual A+0 = A.

What is the absorption law?

A + A·B = A (and its dual A·(A + B) = A). A term absorbs any product that already contains it, because whenever A·B is true A is already true, so the extra term cannot change the result. It is one of the most useful simplification levers.

These laws are the hand tools; a Karnaugh map and Quine-McCluskey automate the same search for the minimal form, and one specific pattern they lean on, keeping a redundant term to kill a glitch, is the consensus theorem.

Every lesson here builds toward one thing: a working CPU, from the transistor up.

Open the free lab →

Builds towardThe consensus theorem