// laws and rules · Logical equivalence
De Morgan's law (disjunction)_
The second De Morgan law says that negating “p or q” is equivalent to asserting “not p and not q”: ¬(p ∨ q) ⇔ (¬p ∧ ¬q). It captures the only way a disjunction can be false: both parts must be false. Like the first law it is a tautology and an essential tool for simplifying negated conditions.
Example
¬(p ∨ q) ⇔ (¬p ∧ ¬q)
What the variables mean
- ▸ p: “The file is empty”
- ▸ q: “The file is corrupted”
In plain words
“The file is neither empty nor corrupted” is equivalent to “it is not the case that it is empty or corrupted”.
Truth table
| p | q | p ∨ q | ¬(p ∨ q) | ¬p | ¬q | ¬p ∧ ¬q | ¬(p ∨ q) ⇔ (¬p ∧ ¬q)★ |
|---|---|---|---|---|---|---|---|
| T | T | T | F | F | F | F | T |
| T | F | T | F | F | T | F | T |
| F | T | T | F | T | F | F | T |
| F | F | F | T | T | T | T | T |
Classification: Tautology · 4 rows
Statement
¬(p ∨ q) ⇔ (¬p ∧ ¬q). Distributing the negation turns the disjunction into a conjunction and negates each part.
Natural language expresses it with “neither… nor…”: “neither empty nor corrupted” is a conjunction of two negations.
Why it is a tautology: reading the table
Row p = F, q = F: p ∨ q is F and its negation T; ¬p and ¬q are T, and T ∧ T is T. They agree. It is the only row where both sides are true, reflecting that a disjunction fails only when both parts fail.
In the other three rows at least one variable is T, so p ∨ q is T and its negation F; at the same time one of ¬p or ¬q is F and the conjunction is F. Both columns agree in all four rows.
How it is used in proofs
To prove “neither A nor B” it suffices to negate the disjunction A ∨ B, and vice versa. This is common when showing an element is not in the union of two sets.
Together with the first law it removes all negations from compound formulas, leaving them only on variables (negation normal form).
Examples
Programming: `!(x < 0 || x > 100)` is equivalent to `x >= 0 && x <= 100`. The second form directly says “x is in range”.
Everyday: “I won't go out if it rains or it is cold” is equivalent to “I only go out if it does not rain and it is not cold”.
Sets: the complement of A ∪ B is Aᶜ ∩ Bᶜ, the same law applied to membership.
Relation to other laws
It is the dual of the first De Morgan law; each is obtained from the other by replacing every variable with its negation and applying double negation.
It describes the NOR operator: p ↓ q is ¬(p ∨ q), i.e. ¬p ∧ ¬q.
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
How do I remember which law is which? ▼
When the negation goes inside the parentheses the operator “flips”: ∧ becomes ∨ and ∨ becomes ∧. Both laws follow the same recipe.
What does it have to do with NOR? ▼
NOR is exactly the negation of a disjunction, so p ↓ q ⇔ ¬p ∧ ¬q is this very law in different notation.
Does it hold for exclusive or? ▼
Not directly. ¬(p ⊕ q) is equivalent to p ⇔ q, not to ¬p ∧ ¬q. De Morgan's laws are for ∨ and ∧.