Mathematical Logic: Statements and Logical Operations
1. Propositional Logic and Statements
A statement (or proposition) is a declarative sentence that is either true or false, but not both. Statements form the foundation of mathematical logic and reasoning.
Definition: A proposition p is a statement that has exactly one truth value: either True (T, 1) or False (F, 0).
Examples of statements:
- "5 is a prime number" (True)
- "2 + 2 = 5" (False)
- "For all real numbers x, x² ≥ 0" (True)
Non-statements (not propositions):
- "Is it raining?" (question)
- "Close the door!" (command)
- "x + 3 = 7" (open sentence, truth depends on x)
2. Fundamental Logical Operations
Negation (¬, ~)
The negation of a proposition p, denoted ¬p or ~p, has the opposite truth value of p.
| p | ¬p |
|---|---|
| T | F |
| F | T |
Conjunction (∧)
The conjunction p ∧ q ("p AND q") is true only when both p and q are true.
| p | q | p ∧ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Disjunction (∨)
The disjunction p ∨ q ("p OR q") is true when at least one of p or q is true.
| p | q | p ∨ q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
