Exam Summary: Logic and Predicate Calculus
1. Tautology and Logical Equivalence
- Tautology: A statement that is always true.
- Logical Equivalence: Two statements with the same truth values.
- Important rules: Double Negation, Identity, De Morgan's Laws.
2. Satisfiability and Validity
- Validity: Always true in all cases.
- Satisfiability: True in at least one case.
- Used in AI, circuit design, and program verification.
3. Predicates and Predicate Logic
- Predicates become propositions when given a value.
- Example: "x is even" (becomes true when x=2, false when x=3).
4. Quantifiers
- Universal (For all x): Applies to all elements.
- Existential (There exists x): At least one element satisfies the condition.
- Order of quantifiers matters.
5. Negation of Quantifiers
- NOT There exists x P(x) -> For all x NOT P(x) (Negate existential -> universal)
- NOT For all x P(x) -> There exists x NOT P(x) (Negate universal -> existential)
6. Order of Quantifiers
- For all xThere exists y (x + y = 10) -> True (For all x, there exists y).
- There exists yFor all x (x + y = 10) -> False (A single y works for all x).
7. Infinite Unions and Intersections
- Union: Elements appearing in at least one subset.
- Intersection: Elements common to all subsets.