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