Axiomatic Definition of Boolean Algebra
The handful of base rules (Huntington postulates) every Boolean theorem is built on.
Description
Boolean algebra is defined axiomatically by Huntington's postulates: a set with two operations that are closed, commutative, distributive, have identity elements (0 for +, 1 for ·), and complements. Everything else — every theorem you use to simplify logic — is derived from these axioms.
- Closure under + and ·.
- Identity: x + 0 = x and x · 1 = x.
- Commutative: x + y = y + x; x · y = y · x.
- Distributive: x(y + z) = xy + xz and x + yz = (x+y)(x+z).
- Complement: x + x′ = 1 and x · x′ = 0.
- Every axiom comes in a dual pair (swap + ↔ · and 0 ↔ 1).
- So every theorem proved has a dual theorem for free.
- There exist at least two distinct elements (0 ≠ 1).
- Two-valued Boolean algebra is the special case used for circuits.
- Associativity is a theorem, not an axiom (derivable).
At a glance
What
A set {0,1} with operators +, ·, ′ satisfying Huntington's postulates.
Why
A minimal axiom base makes the whole theory rigorous and provable.
How
State closure, commutativity, distributivity, identities, and complements; derive the rest.
Where
The foundation under every Boolean theorem and simplification.
When
Whenever a result must be proved, not just asserted.
Think of it like…
Axioms are the seed crystals; every theorem you later use grows from these few rules.
The postulates
- Closure under + and ·.
- Identity: x + 0 = x and x · 1 = x.
- Commutative: x + y = y + x; x · y = y · x.
- Distributive: x(y + z) = xy + xz and x + yz = (x+y)(x+z).
- Complement: x + x′ = 1 and x · x′ = 0.
Duality
- Every axiom comes in a dual pair (swap + ↔ · and 0 ↔ 1).
- So every theorem proved has a dual theorem for free.
- There exist at least two distinct elements (0 ≠ 1).
- Two-valued Boolean algebra is the special case used for circuits.
- Associativity is a theorem, not an axiom (derivable).
Postulate pairs (dual)
| + form | · form |
|---|---|
| x + 0 = x | x · 1 = x |
| x + x′ = 1 | x · x′ = 0 |
| x + y = y + x | x · y = y · x |
| x+(yz)=(x+y)(x+z) | x(y+z)=xy+xz |
Real-world applications
The 5 Whys
- 1
Why axioms? A minimal rigorous starting point.
- 2
Why duality? Halves the work — one proof, two theorems.
- 3
Why identities/complements? They define how 0,1 and negation behave.
- 4
Why two-valued? Matches hardware logic levels.
- 5
Root cause: a small consistent axiom set makes all of Boolean algebra provable.
Cheat sheet
Working principle
- State closure, commutativity, distributivity, identities, and complements; derive the rest.
- A set {0,1} with operators +, ·, ′ satisfying Huntington's postulates.
Formulas & Boolean expressions
- x + 0 = x
- x · 1 = x
- x + x′ = 1
- x · x′ = 0
- Identity: x + 0 = x and x · 1 = x.
- Commutative: x + y = y + x; x · y = y · x.
- Distributive: x(y + z) = xy + xz and x + yz = (x+y)(x+z).
- Complement: x + x′ = 1 and x · x′ = 0.
Key facts
- Closure under + and ·.
- Every axiom comes in a dual pair (swap + ↔ · and 0 ↔ 1).
Why it exists
- Root cause: a small consistent axiom set makes all of Boolean algebra provable.