1. Idea of the Connective OR
The connective OR is used to combine two statements when at least one of them needs to be true. In mathematics, OR is usually interpreted in the inclusive sense, meaning one or the other or both.
This connective allows situations where more than one possibility may satisfy a condition.
1.1. Why the inclusive OR is used
Mathematics treats OR inclusively because it keeps reasoning consistent. When a statement says "P or Q", it includes the case when both P and Q are true.
2. Formal Meaning of OR
If P and Q are statements, then the compound statement:
\(P \text{ or } Q\)
is true whenever at least one of P or Q is true.
2.1. Understanding the structure
OR widens the possibilities: even if one statement fails, the overall statement may still hold as long as the other is true.
3. Truth Behaviour of OR
The OR connective has a simple truth behaviour where only one combination gives a false result.
3.1. Intuitive truth patterns
- P true, Q true → true
- P true, Q false → true
- P false, Q true → true
- P false, Q false → false
4. Examples
These examples show how OR works in real situations and in mathematics.
4.1. Everyday examples
- "You can take the bus or you can walk."
- "The device can run on battery or on power supply."
In everyday use, OR may sound exclusive, but logically both options may be true.
4.2. Mathematical examples
- P: "n is even"
- Q: "n is a multiple of 3"
- Compound: "n is even or n is a multiple of 3"
This holds for any number that satisfies at least one of the two conditions.
5. Notes and Observations
Important points about the connective OR:
- Mathematical OR is inclusive, not exclusive.
- Only one case makes OR false: when both parts are false.
- OR expands possibilities — it is more flexible than AND.
- Many mathematical definitions and proofs use OR to allow multiple valid cases.