Textbook
For propositions $P$ and $Q$, the disjunction $P\lor Q$ is true when at least one is true. It is false only when both are false. Mathematical “or” includes the case where both conditions hold.
For example, 4 satisfies “even or greater than 3” for both reasons. This is not exclusive or: an extra successful condition does not cancel truth. Like other logical connectives, disjunction describes a truth value rather than a required order for executing checks in a program.
To prove a particular disjunction, establishing one true part is sufficient. To refute it, both parts must be false. Exchanging the parts preserves the result: $P\lor Q$ and $Q\lor P$ have identical truth values. Repeating a condition adds nothing either: $P\lor P$ is true exactly when $P$ itself is true.
Definitions and further reading: Mathematics for Computer Science.
In plain language
The rule “ticket or invitation” allows entry with either document and still allows it with both. Two grounds produce one true answer rather than a penalty for excessive preparation. Unlike an everyday choice of “tea or coffee,” the symbol $\lor$ does not itself require exactly one option. That is the stated rule’s meaning; mathematical “or” cannot answer for sudden changes to the guard’s instructions.
Example
The example compares three combinations of the illustrative conditions. The final step makes both conditions true and confirms that the result remains true. In the formulas, 1 denotes true and 0 denotes false.
Disjunction: At Least One Condition Is Enough
Neither
Two false conditions give false.
Step 1 of 3: Neither
Read all steps
Neither
Two false conditions give false.
- State
- P: falsecurrentQ: falsecurrentfalsecurrent
One holds
One true condition is sufficient.
- State
- P: truecurrentQ: falsecurrenttruecurrent
Both hold
Both are true; inclusive or is also true.
- State
- P: truecurrentQ: truecurrenttruecurrent
The second step shows “true, false.” Swapping the answers would preserve the result: an invitation is enough without a ticket too. The final step distinguishes inclusive or from an exclusive choice: two true conditions preserve admission. Refusal requires removing both grounds. Treating two successes as false changes the rule itself rather than clarifying the disjunction shown here.
Related concepts: Conjunction.