Mathwords logoMathwords

Inclusive or — Definition, Meaning & Examples

Inclusive or

A disjunction for which either or both statements may be true.

For example, the use of the word or in "A triangle can be defined as a polygon with three sides or as a polygon with three vertices" is inclusive. Either or both options can be true.

Note: In math, or is understood to be inclusive unless stated otherwise.

See also

Exclusive or

Key Formula

P∨QP \lor Q
Where:
  • PP = The first statement (proposition)
  • QQ = The second statement (proposition)
  • ∨\lor = The logical 'or' operator (disjunction), true when at least one of P or Q is true

Example

Problem: Determine the truth value of the statement: 'x is even or x is greater than 3' for x = 4, x = 5, x = 2, and x = 1.
Step 1: Let P = 'x is even' and Q = 'x is greater than 3.' With inclusive or, the compound statement P ∨ Q is true whenever at least one of P or Q is true.
Step 2: For x = 4: P is true (4 is even) and Q is true (4 > 3). Both are true, so the inclusive or is true.
P=T,  Q=T  ⇒  P∨Q=TP = T,\; Q = T \;\Rightarrow\; P \lor Q = T
Step 3: For x = 5: P is false (5 is odd) and Q is true (5 > 3). One is true, so the inclusive or is true.
P=F,  Q=T  ⇒  P∨Q=TP = F,\; Q = T \;\Rightarrow\; P \lor Q = T
Step 4: For x = 2: P is true (2 is even) and Q is false (2 is not greater than 3). One is true, so the inclusive or is true.
P=T,  Q=F  ⇒  P∨Q=TP = T,\; Q = F \;\Rightarrow\; P \lor Q = T
Step 5: For x = 1: P is false (1 is odd) and Q is false (1 is not greater than 3). Neither is true, so the inclusive or is false.
P=F,  Q=F  ⇒  P∨Q=FP = F,\; Q = F \;\Rightarrow\; P \lor Q = F
Answer: The statement is true for x = 4, x = 5, and x = 2, and false only for x = 1. The inclusive or is false only when both individual statements are false.

Frequently Asked Questions

What is the difference between inclusive or and exclusive or?
Inclusive or is true when one or both statements are true—it is only false when both are false. Exclusive or is true when exactly one statement is true but not both. For example, 'You can have soup or salad' in a restaurant usually means exclusive or (pick one), while 'Students who study math or science may apply' uses inclusive or (studying both still qualifies you).
Why does math always use inclusive or?
Mathematics defaults to inclusive or because it is the more general and logically useful interpretation. If a theorem says a result holds when condition A or condition B is met, it would be unnecessarily restrictive to exclude cases where both conditions happen to be true. When mathematicians need the exclusive sense, they say 'either A or B, but not both' explicitly.

Inclusive or vs. Exclusive or

Inclusive or (P∨QP \lor Q) is true when P is true, Q is true, or both are true—it is false only when both are false. Exclusive or (P⊕QP \oplus Q) is true only when exactly one of P or Q is true, and false when both are true or both are false. In everyday English, 'or' is often exclusive ('tea or coffee?'), but in mathematics, 'or' always means inclusive or unless otherwise specified.

Why It Matters

Inclusive or is the standard meaning of 'or' across all of mathematics, from set theory (union) to logic proofs to probability. When you see 'A or B' in a math problem, theorem, or definition, you should assume both can be true simultaneously. Misreading it as exclusive or can lead to incorrect solutions, especially in counting problems and probability where overlapping cases must be included.

Common Mistakes

Mistake: Assuming 'or' in math means 'one or the other but not both,' based on everyday English usage.
Correction: In mathematics, 'or' is always inclusive. The statement 'A or B' is true even when both A and B are true. If exclusivity is intended, it will be stated explicitly as 'A or B, but not both.'
Mistake: Marking an inclusive or statement as false when both components are true.
Correction: The only way an inclusive or statement is false is when both components are false. If both are true, the disjunction is still true—that is precisely what makes it 'inclusive.'

Related Terms

  • Exclusive or — Or that excludes the both-true case
  • Disjunction — Formal name for an or statement
  • Conjunction — Logical and, the companion connective
  • Truth Table — Table showing all truth value combinations
  • Negation — Logical not, another basic connective
  • Union — Set operation that mirrors inclusive or