// laws and rules · Logical equivalence
Distributive law of ∨ over ∧_
This law is the twin of the distributive law of ∧ over ∨, and it is surprising because arithmetic has no counterpart: a + (b · c) is not equal to (a + b) · (a + c). In logic, however, both directions are valid and each is proved with an eight-row table.
Example
(p ∨ (q ∧ r)) ⇔ ((p ∨ q) ∧ (p ∨ r))
What the variables mean
- ▸ p: “I have a scholarship”
- ▸ q: “I pass the exam”
- ▸ r: “I hand in the project”
In plain words
“I have a scholarship, or else (I pass the exam and hand in the project)” is equivalent to “(I have a scholarship or I pass the exam) and (I have a scholarship or I hand in the project)”.
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 | F | T | T | T | T | T |
| T | F | T | F | T | T | T | T | T |
| T | F | F | F | T | T | T | T | T |
| F | T | T | T | T | T | T | T | T |
| F | T | F | F | F | T | F | F | T |
| F | F | T | F | F | F | T | 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). Like every equivalence, it is proved by checking that the biconditional between both sides is a tautology.
Translated into Boolean-algebra notation it would read p + (q · r) = (p + q) · (p + r), an identity that is false for numbers but true in logic. The reason is that here “addition” is idempotent: p ∨ p is p, whereas p + p is not p.
Why it holds: reading the table
If p is T, the left side is T because a disjunction with one true member is true; and so is the right side, because p appears inside both brackets. All four rows with p = T give T on both sides.
If p is F, the left side reduces to q ∧ r, and the right side also reduces to q ∧ r because each bracket keeps its second member. They agree again, so the final biconditional column is T in all eight rows.
How it is used
Left to right it pushes a formula towards conjunctive normal form (a conjunction of disjunctions), the format consumed by SAT solvers and by the resolution method.
Right to left it factors: when the same term is repeated across several alternatives in a condition, you can pull it out and write it once.
Examples
Everyday: “It is a holiday, or (it is Saturday and I am not working)” is equivalent to “(it is a holiday or it is Saturday) and (it is a holiday or I am not working)”.
Programming: `isAdmin || (active && verified)` is equivalent to `(isAdmin || active) && (isAdmin || verified)`. The short form is usually better here, but the long one is what tools requiring conjunctive normal form expect.
Relation to other laws
Together with the distributive law of ∧ over ∨ it forms a symmetric pair. That symmetry between ∧ and ∨ is an instance of the duality principle: every law stays valid when you swap ∧ with ∨ and T with F.
Applying De Morgan to this law yields the other distributive law, so proving one of the two is enough to have both.
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 it fail in arithmetic? ▼
Because numeric addition is neither idempotent nor absorbing: 2 + (3 · 4) = 14, but (2 + 3) · (2 + 4) = 30. In logic p ∨ p = p, and that is what makes the law valid.
Which of the two distributive laws should I use? ▼
It depends on your goal: ∨ over ∧ leads to conjunctive normal form; ∧ over ∨ leads to disjunctive normal form.
What is the duality principle? ▼
It says that swapping ∧ for ∨ and true for false in a valid law produces another valid law. These two distributive laws are duals of each other.