Sheet2 - McMaster University Department of Computing and...

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McMaster University Department of Computing and Software Dr. W. Kahl COMP SCI 1FC3 Exercise Sheet 2 COMP SCI 1FC3 — Mathematics for Computing 16 January 2009 Exercise 2.1 — Predicates, formal and English From the textbook section 1.3, do at least exercises 10–14, 25, 28, 30. 38–42 (pp. 47–49). Exercise 2.2 — Predicate Logic Equivalences Using logical equivalences from the text book, prove the following: (a) (∃ x ( P ( x ))) ∨ (∃ x ( Q ( x ))) ≡ ∃ x ( P ( x ) ∨ Q ( x )) Using semantic arguments (and possibly logical equivalences), prove the following (assuming x does not occur free in A , and that the domain of quanti±cation is non-empty): (b) (∀ x ( P ( x ))) ∧ A ≡ ∀ x ( P ( x ) ∧ A ) (c) (∃ x ( P ( x ))) ∧ A ≡ ∃ x ( P ( x ) ∧ A ) (d) x ( A P ( x )) ≡ A → ∀ x ( P ( x )) (e) x ( A P ( x )) ≡ A → ∃ x ( P ( x )) (f) x ( P ( x ) → A ) ≡ (∀ x ( P ( x ))) → A ) (g) x ( P ( x ) → A ) ≡ (∃ x ( P ( x ))) → A
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This note was uploaded on 03/26/2011 for the course CS 1fc3 taught by Professor Kahl during the Spring '11 term at McMaster University.

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