Boolean Algebra · step 1 of 13

Boolean axioms & laws

A Boolean algebra is a set with two operations + and ·, a complement ′, and constants 0 and 1 satisfying a short axiom list: commutativity, associativity, distributivity BOTH ways, identity (x + 0 = x, x·1 = x) and complement (x + x′ = 1, x·x′ = 0). Everything else — absorption, idempotence, De Morgan — is derivable, and exam questions want the derivation, not the recollection.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Know which laws are axiomsOnly the axioms may be used without proof. Absorption and De Morgan are theorems, and some courses require you to prove them before use.
Note both distributive laws holdx(y + z) = xy + xz AND x + yz = (x + y)(x + z). The second has no arithmetic analogue and is the one people forget.
Apply idempotence and dominationx + x = x, x·x = x, x + 1 = 1, x·0 = 0. These collapse expressions fast.
Justify every step by nameWrite the law beside each line. Marks are attached to the justifications, not the final expression.

Worked example

Prove the absorption law x + xy = x from the axioms.

x + xy = x·1 + xy, by the identity law.
= x(1 + y), by distributivity.
= x·1, since 1 + y = 1 by domination.
= x, by identity.

Answer. x + xy = x, derived in four axiom-justified steps.

Where marks get dropped

These are the specific errors that cost credit on boolean axioms & laws questions — QED's rubric penalises each of them separately.

Carrying arithmetic intuition across. In Boolean algebra 1 + 1 = 1, and x + yz factors as (x + y)(x + z) — neither holds for numbers.
Using absorption or De Morgan as an axiom when the question asks you to derive them.
Dropping the justification column. An unjustified simplification chain typically scores zero for method.

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Marked like an examiner

Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.

Answer in real notation

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Boolean axioms & laws — frequently asked questions

Why does x + 1 = 1?

Because + is logical OR: if one input is true the result is true regardless. Formally it follows from the complement and distributive axioms, and it is the domination law.

Are the axioms independent?

Huntington’s postulates are a minimal set, and it is a classic exercise to show none can be dropped. Most textbooks list a redundant but convenient collection.

Is Boolean algebra the same as propositional logic?

They are the same structure with different notation: + is ∨, · is ∧, ′ is ¬, and 1 and 0 are ⊤ and ⊥. Set algebra is a third notation for the same thing.

The rest of Boolean Algebra

Axioms, laws, simplification and Boolean functions. Each subtopic below has its own method, worked example and mark-losing traps.

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