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x1-sample-slides - Problem 1 Are and logically equivalent...

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Problem 1. Are ¬ ( ) and ¬ ↔ ¬ logically equivalent? Justify your answer. No. For example if and are both assigned T, then ¬ ( ) is F and ¬ ↔ ¬ is T. In fact ¬ ↔ ¬ is logically equivalent to . So under any assignment of truth values to and these two sentential formulas get opposite truth values. 1
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Problem 2. Indicate which of the following assertions of logical equivalence are true and which are false. (a) ( P ( ) Q ( ) ) ≡ ∃ P ( ) ∨ ∃ Q ( ) is true. (b) ( P ( ) Q ( ) ) ≡ ∃ P ( ) ∧ ∃ Q ( ) is false. (c) ( P ( ) Q ( ) ) ≡ ∀ P ( ) → ∀ Q ( ) is false. (d) ( P ( ) Q ( ) ) ≡ ∀ P ( ) ↔ ∀ Q ( ) is false. 2
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Problem 3. Let the domain of discourse be the real numbers. Determine whether each of the following is true or false. (a) ∀ ∃ 2 = 2 is true: Let be the same as . (b) ∀ ∃ = 2 is false: If is negative, it has no real square root. (c) ∃ ∀ = 0 is true: Consider = 0. (d) ∀ ∃ + = 1 is true: Let be 1 - . 3
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Problem 4. Find a formula that is logically equivalent to ¬∀ ∃ ( P ( ) → ∃ R ( ) ) in which any negation symbols immediate precede predicate (relation) sym- bols.
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