09ma6cHW2

09ma6cHW2 - Ma/CS 6c Assignment #2 Due Wednesday, April 14,...

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Unformatted text preview: Ma/CS 6c Assignment #2 Due Wednesday, April 14, at 1 p.m. (30%) 1. (i) Let A be a wff and assume that the only connectives appearing in A are among ¬, ∧, ∨ (i.e., ⇒, ⇔ don’t appear). Let A∗ be obtained from A by replacing each propositional variable p appearing in A by ¬p and replacing ∧ by ∨ and ∨ by ∧. Show that ¬A ≡ A∗ (i.e., ¬A, A∗ are logically equivalent). (ii) Suppose A is a wff as in (i). Let A be the wff obtained from A by replacing ∧ by ∨ and ∨ by ∧. We call A the dual of A. (Example: (p ∨ q ) ∧ ¬r is the dual of (p ∧ q ) ∨ ¬r.) Show that A is a tautology iff ¬A is a tautology. (iii) (Principle of duality) For A, B wff as in (ii), show that A ≡ B iff A ≡ B . (20%) 2. Consider the wff An = ((. . . ((p1 ⇔ p2 ) ⇔ p3 ) ⇔ . . . ) ⇔ pn ). Show that a valuation v satisfies An exactly when v (pi ) = 0 for an even number of i in the interval 1 ≤ i ≤ n. (20%) 3. For each n = 2, 3, 4, . . . find a set S = {A1 , A2 , . . . , An } consisting of n wff such that S is not satisfiable, but any proper nonempty subset S S is satisfiable. (30%) 4.∗ A set S of wff is independent if for any wff A ∈ S , S \ {A} |= A, i.e., A is not implied logically by the rest of the wff in S . (So, by definition, the empty set ∅ is independent, and S = {A} is independent iff A is not a tautology.) (i) Which of the sets (a) {p ⇒ q, q ⇒ r, r ⇒ q } (b) {p ⇒ q, q ⇒ r, p ⇒ r} (c) {p ⇒ r, r ⇒ q, q ⇒ p, r ⇒ (q ⇒ p)} are independent and which are not? (ii) Two sets of wff, S, S are called equivalent if S |= A for any A ∈ S and S |= A for any A ∈ S . (So, by definition, if S = {A}, where A is a tautology, ∅ is equivalent to S .) Show that for any finite set S of wff, there is a subset S ⊆ S which is independent and equivalent to S . 1 ...
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