Disjunction — Definition, Examples & Truth Table
Disjunction
A statement which connects two other statements using the word or.
For example, "A polygon with four sides can be called a quadrilateral or a quadrangle" contains the disjunction "quadrilateral or quadrangle".
See also
Key Formula
p∨q
Where:
- p = The first statement (called a disjunct)
- q = The second statement (called a disjunct)
- ∨ = The logical "or" operator, read as "or"
Example
Problem: Let p be the statement "7 is odd" and let q be the statement "7 is prime." Determine the truth value of the disjunction p ∨ q.
Step 1: Evaluate statement p. Is 7 odd? Yes — 7 is not divisible by 2, so p is TRUE.
p=TRUE
Step 2: Evaluate statement q. Is 7 prime? Yes — 7 has no divisors other than 1 and itself, so q is TRUE.
q=TRUE
Step 3: Apply the rule for disjunction: p ∨ q is true when at least one of p or q is true. Here both are true.
p∨q=TRUE∨TRUE=TRUE
Answer: The disjunction "7 is odd or 7 is prime" is TRUE. Note that it would also be true if only one of the two statements were true. The only way a disjunction is false is if both p and q are false.
Another Example
Problem: Let p be the statement "10 is odd" and let q be the statement "10 is negative." Find the truth value of p ∨ q.
Step 1: Evaluate statement p. Is 10 odd? No — 10 is divisible by 2, so p is FALSE.
p=FALSE
Step 2: Evaluate statement q. Is 10 negative? No — 10 is positive, so q is FALSE.
q=FALSE
Step 3: A disjunction is false only when both disjuncts are false. Since both p and q are false, the disjunction is false.
p∨q=FALSE∨FALSE=FALSE
Answer: The disjunction "10 is odd or 10 is negative" is FALSE, because neither component statement is true.
Frequently Asked Questions
What is the difference between disjunction and conjunction in math?
A disjunction (p ∨ q) uses "or" and is true when at least one statement is true. A conjunction (p ∧ q) uses "and" and is true only when both statements are true. So disjunction is the more lenient condition — only one part needs to hold.
Is disjunction inclusive or exclusive?
In standard math and logic, disjunction (∨) is inclusive: p ∨ q is true when p is true, when q is true, or when both are true. This differs from everyday English, where "or" sometimes implies "one or the other but not both" (exclusive or). Exclusive or is a separate operation, written p ⊕ q.
Disjunction (OR) vs. Conjunction (AND)
A disjunction p ∨ q requires at least one of the two statements to be true. A conjunction p ∧ q requires both statements to be true. The disjunction is false in only one row of the truth table (both false), while the conjunction is true in only one row (both true).
Why It Matters
Disjunction is one of the fundamental operations in logic and appears throughout mathematics, computer science, and everyday reasoning. In programming, the OR operator mirrors disjunction and controls how conditions are evaluated. Understanding when a disjunction is true or false is essential for working with truth tables, logical proofs, and set theory (where disjunction corresponds to the union of sets).
Common Mistakes
Mistake: Assuming "or" means "one or the other but not both" (exclusive or).
Correction: In mathematics, disjunction is inclusive by default. p ∨ q is true even when both p and q are true. If you need "one but not both," you must use exclusive or (⊕) explicitly.
Mistake: Thinking a disjunction is false when only one statement is true.
Correction: A disjunction is false only when both component statements are false. If even one disjunct is true, the entire disjunction is true.
Related Terms
- Conjunction — Logical AND — the counterpart to OR
- Inclusive Or — The standard meaning of disjunction
- Exclusive Or — OR but not both — different from disjunction
- Negation — Logical NOT — another basic logical operation
- Truth Table — Tool used to evaluate disjunctions
- Quadrilateral — Used in the classic disjunction example