Laws of Boolean algebra

Also called: Boolean laws, Boolean identities, Boolean theorems, Boolean postulates, Boolean axioms, laws

The equations that hold for every value of their variables, such as A + 0 = A or A + AB = A, used to rewrite and simplify expressions.

The laws of Boolean algebra are equations that are true for every possible input. Because each side always equals the other, you may replace one with the other at any step. That lets you simplify an expression without writing out its whole truth table, which matters once there are 16, 64 or more rows.

The usual list, each with an OR form and an AND form:

Identity: = A, = A
Null: = 1, = 0
Idempotent: = A, = A
Complement: = 1, = 0
Involution: = A
Commutative, associative and distributive
Absorption: = A
De Morgan: =
Consensus: =

The pairs exist because of the duality principle. Every letter can stand for a whole expression; see substitution rule. Write the law's name next to each step of a simplification so it can be checked.

Worked example

Example

Naming every step

Simplify , naming the law at each step.

1.
= 1 (complement), giving .
2.
= B (identity) and = 0 (null), giving .
3.
= B (identity).
4.
Result: B. A never mattered.

Common mistakes

Using a rule from ordinary algebra that is not a Boolean law, such as cancelling or dividing.
Using only one form of a law. Every law has a dual, and you may need either.
Trusting a half-remembered law. Check it with a small truth table first.

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Laws of Boolean algebra is taught in Boolean Algebra.

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