Distributive law: A AND (B OR C) = (A AND B) OR (A AND C), both forms
The distributive law says that AND distributes over OR and OR distributes over AND: A AND (B OR C) = (A AND B) OR (A AND C) and A OR (B AND C) = (A OR B) AND (A OR C).
OR distributes over AND (and AND over OR), turning one grouping into a product of sums or a sum of products.
The first form matches ordinary algebra (a product distributes over a sum). The second has no counterpart with numbers: a + bc is not (a + b)(a + c). In Boolean algebra both hold, and that symmetry is what makes the sum-of-products and product-of-sums forms interchangeable.
Distribution is the engine of normal forms. Applying the first form repeatedly turns any AND-OR expression into a sum of products (DNF); applying the second turns it into a product of sums (CNF). It is also the step that can make a formula grow: distributing an OR over k ANDs of n literals produces up to n to the power k clauses, which is why the normal-form converter has a size bound and reports it.
The eight-row proof over A, B and C is below, and the worked example compares the factored and expanded forms row by row.
Statement
A OR (B AND C) = (A OR B) AND (A OR C) Proof by truth table: A OR (B AND C) = (A OR B) AND (A OR C)
| # | A | B | C | lhs | rhs |
|---|---|---|---|---|---|
| 0 | FALSE | FALSE | FALSE | FALSE | FALSE |
| 1 | FALSE | FALSE | TRUE | FALSE | FALSE |
| 2 | FALSE | TRUE | FALSE | FALSE | FALSE |
| 3 | FALSE | TRUE | TRUE | TRUE | TRUE |
| 4 | TRUE | FALSE | FALSE | TRUE | TRUE |
| 5 | TRUE | FALSE | TRUE | TRUE | TRUE |
| 6 | TRUE | TRUE | FALSE | TRUE | TRUE |
| 7 | TRUE | TRUE | TRUE | TRUE | TRUE |
The same law in four notations
A OR (B AND C)A || (B && C)A ∨ (B ∧ C)A + (B·C)A + \left(B \cdot C\right)(A OR B) AND (A OR C)(A || B) && (A || C)(A ∨ B) ∧ (A ∨ C)(A + B)·(A + C)\left(A + B\right) \cdot \left(A + C\right)The common slip
Factoring where nothing is shared. (A AND B) OR (C AND D) has no common factor, and “distributing” it back produces a product of four sums, not a shorter formula; the simplifier keeps the shorter of the two forms.
Questions
Which direction does the simplifier use?
The simplifier neither expands nor factors on its own: it applies only laws that shorten an expression, and distribution is not one of them. The normal-form converter uses the law in whichever direction the target form needs, so the CNF of (A AND B) OR (A AND C) is A AND (B OR C) and the DNF of A AND (B OR C) is the expanded form.
Related
Lesson that teaches this
Pitfalls
- “A or B and C”: the grouping ambiguity and how to resolve it
- A timeout is not a pass: “Not determined” results
Tools
- Boolean algebra calculator and simplifier with steps (every law named and verified)
- Logical equivalence checker: two expressions, exact verdict, counterexample
Practice
See also
- Distribution in practice: A AND (B OR C) expanded
- Lesson 10: Normal forms: DNF, CNF, minterms, and maxterms (advanced, 15 min)
- CNF vs DNF: differences, examples and when to use each
- Sum of products vs product of sums
- Absorption law: A OR (A AND B) = A, with proof
Laws
Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build . Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.