1. Idea of the Connective AND
The connective AND is used to join two statements when both of them need to hold at the same time. It creates a single compound statement that is true only if each individual statement is true.
This connective helps describe situations where two conditions must be satisfied together, not separately.
1.1. How AND behaves in reasoning
When statements are joined using AND, the overall truth depends on both parts being correct. If even one part fails, the entire combined statement fails.
2. Formal Meaning of AND
If P and Q are statements, then the compound statement:
\(P \text{ and } Q\)
is true exactly when P is true and Q is true.
2.1. Understanding the structure
Both statements must support each other. If one fails, the AND condition breaks. This is why AND narrows down possibilities—it demands two truths at once.
3. Truth Behaviour of AND
The connective AND has a simple truth pattern. The combined statement fails whenever any part fails.
3.1. Intuitive truth patterns
- P true, Q true → true
- P true, Q false → false
- P false, Q true → false
- P false, Q false → false
4. Examples
These examples show how AND is used in everyday and mathematical settings.
4.1. Everyday examples
- "It is raining and it is cold."
- "The key is in the drawer and the drawer is unlocked."
The combined statement is true only when both parts are true.
4.2. Mathematical examples
- P: "n is even"
- Q: "n is a multiple of 4"
- Compound: "n is even and n is a multiple of 4"
True only for numbers that satisfy both conditions at once.
5. Notes and Observations
Important points about the connective AND:
- The combined statement is the strictest among logical connectors.
- Both components must be true for the conjunction to be true.
- Using AND reduces the number of cases where the statement holds.
- In reasoning, AND is often used to describe two conditions that must occur together.