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# Logic1116 - Correctness for proposifS tional logic 1 2 3 HW...

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LS° °°° °° Correctness for proposi- tional logic 1. HW solutions 2. Correctness 3. Completeness

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HW #8 (due Monday, 11/23) 1. Let A be a formula in the propositional language (no quantifiers). Let cA be the number of places at which a binary connective (one of , v, =>) occurs. Let sA be the number of places at which propositional variables occur. Show that sA = cA+1. 2. Exercise 17.1 3. Exercise 17.2 (a)
HW #7, Q1. (complete induction) P(n): n is a product of primes. For any n (>1), assume P holds for all numbers less than n. We have to show P(n). n is a prime or not. If it is a prime, then P holds. If it is not, there are other factors of n be- sides 1 and n. Let n = ab (1<a, b<n). Since a and b are smaller than n, P holds for a and b. a and b are products of primes, and n is a product of products of

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(finite descent) Suppose n (>1) is not a product of primes. Then n is not a prime and n = ab (1<a, b<n). If each of a and b is a product of primes, n is a product of primes, which it is not. Therefore, a or b is not a product of primes. Since for every n, ~P(n) => ~P(m) for some m<n, P holds for all n.
Q2. show 15.4 => 15.1 Assume 15.4 and 15.1 (1) & (2): P(0) and for every n, P(n) => P(n+1). Show that P holds for every natural number. (proof by contradiction) assume P fails for some n (≠0). Then for every such n, P fails for n-1 by the contrapositive of (2). Thus for every n, ~P(n) => ~P(n-1). Then by 15.4, P holds for all n, which is a

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16.15 (compactness) A universe V with denumerably many people There is at least one club.
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