// reference · OR
∨ Disjunction_
Disjunction joins two propositions with the word "or". The expression p ∨ q is true when at least one of them is true — including when both are. It is false only when p and q are both false. This inclusive "or" is the default meaning in mathematics.
Truth table: p ∨ q
| p | q | p ∨ q★ |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Classification: Contingency · 4 rows
Definition
The disjunction of p and q, written p ∨ q, is false only when both propositions are false. In the remaining three cases it is true. The symbol ∨ comes from the Latin vel (inclusive "or"), as opposed to aut (exclusive "or").
Alternative notations: p | q or p || q (programming), p + q (Boolean algebra), and p OR q (electronics). The calculator accepts ∨, | and +.
How to read it
p ∨ q is read "p or q" and, when precision matters, "p or q, or both". For instance, "you can pay by card or in cash" formalizes as p ∨ q, and nothing stops you from paying part with each.
Everyday English often uses an exclusive "or" ("tea or coffee?"). In logic, if you mean exactly one of the two, use exclusive disjunction ⊕, not ∨.
When it is true
The table has four rows. With p = T and q = T, p ∨ q = T. With p = T and q = F, p ∨ q = T. With p = F and q = T, p ∨ q = T. With p = F and q = F, p ∨ q = F.
Three true rows and one false row: p ∨ q is a contingency. Think of adding zeros and ones capped at 1: 0 + 0 = 0 and every other sum gives 1.
Everyday example
A discount applies if "you are a student or you are over 65". A student gets it, a 70-year-old gets it, and a 70-year-old student gets it too. Only someone who meets neither condition is left out.
In programming, if (isAdmin || isOwner) grants access when either condition holds.
Properties and equivalences
Commutative: p ∨ q ≡ q ∨ p. Associative: (p ∨ q) ∨ r ≡ p ∨ (q ∨ r). Idempotent: p ∨ p ≡ p. Identity element: p ∨ 0 ≡ p. Annihilator: p ∨ 1 ≡ 1.
Distributive over conjunction: p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r). Absorption: p ∨ (p ∧ q) ≡ p. Excluded middle: p ∨ ¬p is a tautology.
Equivalences with other operators: p ∨ q ≡ ¬(¬p ∧ ¬q) (De Morgan), p ∨ q ≡ ¬p ⇒ q (material implication), and p ∨ q ≡ ¬(p ↓ q) — disjunction is the negation of NOR.
Common mistakes
Treating ∨ as exclusive. In logic p ∨ q is true when both are true. If a statement says "one or the other, but not both", the right operator is ⊕.
Getting De Morgan wrong: ¬(p ∨ q) is equivalent to ¬p ∧ ¬q ("neither p nor q"), not ¬p ∨ ¬q. And forgetting precedence: ∧ binds tighter than ∨, so p ∨ q ∧ r means p ∨ (q ∧ r).
Equivalent expressions
Try it yourself
Edit the expression in the calculator and watch how every step of the table changes.
Open in the calculator →Related laws and rules
Logical equivalence
De Morgan's law (disjunction)
¬(p ∨ q) ⇔ (¬p ∧ ¬q)
Logical equivalence
Distributive law of ∨ over ∧
(p ∨ (q ∧ r)) ⇔ ((p ∨ q) ∧ (p ∨ r))
Fundamental law
Law of excluded middle
p ∨ ¬p
Rule of inference
Disjunctive syllogism
((p ∨ q) ∧ ¬p) ⇒ q
Logical equivalence
Material implication
(p ⇒ q) ⇔ (¬p ∨ q)
Logical equivalence
Absorption law
(p ∨ (p ∧ q)) ⇔ p
Frequently asked questions
Is p ∨ q true when p and q are both true? ▼
Yes. Logical disjunction is inclusive: it is true when at least one proposition is true, including both at once. It is false only when both are false.
What is the difference between ∨ and ⊕? ▼
∨ is the inclusive "or" (true with one or both). ⊕ is the exclusive "or" (true with exactly one). They differ only in the row p = T, q = T.
How do you negate a disjunction? ▼
With De Morgan's law: ¬(p ∨ q) ≡ ¬p ∧ ¬q. Denying "p or q" is the same as asserting "neither p nor q".