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### solution10_08

Course: COMP 109, Fall 2009
School: East Los Angeles College
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Word Count: 368

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of Foundations Computer Science (COMP109) Exercise 10 1. Explain intuitively the definition of sentences: A formula is a sentence if every occurence of a variable is the scope of either or . Therefore, the truth of a sentence in a model (D, a) does not depend on the assignment a, and we can simply write D |= or D |= . 2. Suppose S and &lt; are binary predicate symbols and 1 is an individual constant....

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of Foundations Computer Science (COMP109) Exercise 10 1. Explain intuitively the definition of sentences: A formula is a sentence if every occurence of a variable is the scope of either or . Therefore, the truth of a sentence in a model (D, a) does not depend on the assignment a, and we can simply write D |= or D |= . 2. Suppose S and < are binary predicate symbols and 1 is an individual constant. Consider the structure N = N, S N , <N , 1 with N = {1, 2, 3, . . .}, S N = {(n, n + 1) | n N}, <N =<, 1 = 1, Which of the following are true? N |= x <(1, x)? True N |= x <(x, 1)? False N |= xy <(x, y)? True N |= xy <(x, y)? False. Set, for example, a(x) = 1 and a(y) = 1 N |= xy <(x, y)? True N |= xy <(y, x)? False. Set a(x) = 1. 1 N N Let a be an assignment into N such that a(x) = 2 and a(y) = 4. Which of the following are true? (N , a) |= <(x, y)? True, because a(x) = 2 < 4 = a(y). (N , a) |= S(x, y)? False, because (2, 4) S N . (N , a) |= S(x, x)? False, because (2, 2) N S . 3. Which of the following are tautologies? Provide structures refuting the non-tautologies. x(P (x) P (x))? Tautology xyQ(x, y)? No tautology. Take, for example, any structure D with QD = . Then D |= xyQ(x, y). x((P1(x) P2 (x)) (P2 (x) P1 (x)))? Tautology P (c) xP (x) (c is a constant.) Tautology P (c) xP (x) (c is a constant.) No tautology. Take, for example, the structure D = ({1, 2}, P D , cD ), where P D = {1} and cD = 1. Then D |= P (c), but D |= xP (x). Therefore, D |= P (c) xP (x). 4. Let be a set of formulas and a formula. Define ` |= ': |= if and only if for every model (D, a): i...

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