990 B
990 B
Propositions and Predicates
Source: MIT 6.1200J Lecture 01
Proposition
Definition: A statement that is either True or False.
Examples
- True: 2 + 3 = 5
- False: 2 + 3 = 6
Non-examples (not propositions)
- "Hello" — not a statement
- "Who are you?" — a question, not a declarative statement
Predicate
Definition: A proposition whose truth depends on variables.
Examples
- P(n) = "n² + n + 41 is prime" where n ∈ ℕ
- Q(x, y) = "x + y = 5" where x, y ∈ ℝ
Notation
- ∀ (for all): ∀n ∈ ℕ. P(n)
- ∃ (there exists): ∃n ∈ ℕ. P(n)
Implication (A ⇒ B)
Key point: A ⇒ B is NOT about causation or time ordering.
Truth table:
| A | B | A ⇒ B |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Note: "If and only if" (A ⟺ B) means (A ⇒ B) AND (B ⇒ A)