// laws and rules · Logical equivalence
Distributive law of ∧ over ∨_
Conjunction distributes over disjunction exactly as multiplication distributes over addition in arithmetic. It lets you factor a formula out or expand it without changing its truth value, and it is one of the basic tools for converting expressions into normal form.
Example
(p ∧ (q ∨ r)) ⇔ ((p ∧ q) ∨ (p ∧ r))
What the variables mean
- ▸ p: “I have time”
- ▸ q: “I go to the cinema”
- ▸ r: “I go to the theatre”
In plain words
“I have time and (I go to the cinema or the theatre)” says exactly the same as “(I have time and I go to the cinema) or (I have time and I go to the theatre)”.
Truth table
| p | q | r | q ∨ r | p ∧ (q ∨ r) | p ∧ q | p ∧ r | (p ∧ q) ∨ (p ∧ r) | (p ∧ (q ∨ r)) ⇔ ((p ∧ q) ∨ (p ∧ r))★ |
|---|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T | T | T |
| T | T | F | T | T | T | F | T | T |
| T | F | T | T | T | F | T | T | T |
| T | F | F | F | F | F | F | F | T |
| F | T | T | T | F | F | F | F | T |
| F | T | F | T | F | F | F | F | T |
| F | F | T | T | F | F | F | F | T |
| F | F | F | F | F | F | F | F | T |
Classification: Tautology · 8 rows
Statement
The law states that p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r). The biconditional between the two sides is a tautology, which is how an equivalence is proved with truth tables.
It is the logical analogue of a · (b + c) = a · b + a · c. In Boolean algebra it is in fact written with that very notation, where ∧ is the product and ∨ the sum.
Why it holds: reading the table
With three variables the table has 2³ = 8 rows. Simply compare the column for p ∧ (q ∨ r) with the one for (p ∧ q) ∨ (p ∧ r): they match row by row, which is why the final biconditional column is T in all eight.
The direct argument: if p is F, both sides are F (a conjunction with a false factor is false). If p is T, the left side is worth whatever q ∨ r is worth, and the right side collapses to q ∨ r because each conjunction keeps its second member. Either way the two sides agree.
How it is used
Applied left to right it expands a formula towards disjunctive normal form (a disjunction of conjunctions), the shape required by many satisfiability algorithms and by circuit synthesis.
Right to left it works as factoring: it reduces the number of operations. In a digital circuit, rewriting (p ∧ q) ∨ (p ∧ r) as p ∧ (q ∨ r) saves one AND gate.
Examples
Everyday: “It is the weekend and (it is raining or it is cold)” is the same as “(it is the weekend and it is raining) or (it is the weekend and it is cold)”.
Programming: the condition `active && (isAdmin || isEditor)` is equivalent to `(active && isAdmin) || (active && isEditor)`. The first form is usually preferable because it evaluates `active` only once.
Relation to other laws
It has a twin, the distributive law of ∨ over ∧, which has no arithmetic counterpart: in logic both directions are valid.
Together with De Morgan's laws and double negation it forms the toolkit used to push any formula into conjunctive or disjunctive normal form.
Try it yourself
Edit the expression in the calculator and watch how every step of the table changes.
Open in the calculator →Related operators
Frequently asked questions
Why does the table have 8 rows? ▼
Because three distinct variables are involved (p, q, r) and the number of rows is always 2ⁿ: 2³ = 8.
Do subtraction or the conditional distribute too? ▼
The conditional does not distribute this way. Convert it to ¬p ∨ q first using material implication, then apply the distributive laws.
Is it useful for simplifying circuits? ▼
Yes. Used right to left it is a factoring step that reduces the gate count, one of the standard moves when minimising Boolean functions.