Strong Disjunctive
Strong Disjunctive (p |v| q)
Read as "either p or q but not both".
The strong disjunctive operator says that if either of the terms (p and q) is true, but not both of them, then the statement is true. This differs from the regular disjunctive in that when both p and q are true then the statement is false. Hence (p v q) ^ ~ (p^q).
| p | q | p | v| q | Comments |
| T | T | F | Both terms are true so the statement is false since the definition of the strong disjunctive says "either, or, but not both." Both terms cannot be true. |
| T | F | T | Only one term is true so the statement is true |
| F | T | T | Only one term is true so the statement is true |
| F | F | F | Neither term is true so the statement is false. One of the terms has to be true. |
Derivation
| p | q | p v q | p ^ q | ~(p ^ q) | (p v q) ^ ~ (p^q) |
| T | T | T | T | F | F |
| T | F | T | F | T | T |
| F | T | T | F | T | T |
| F | F | F | F | T | F |
| (p v q) ^ ~ (p^q) | p | v| q | {p v q) ^ ~ (p^q)} <-> (p | v| q) |
| F | F | T |
| T | T | T |
| T | T | T |
| F | F | T |
Since the result is a tautology our derivation and definition are logically equivalent.
