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def def def in Software Development Code 39 Extended in Software def def def

def def def using barcode generation for software control to generate, create uss code 39 image in software applications. Code 2 of 5 2 = x y ( Software Code 39 P (x, y) P (y, x)) 3 = x y z ((P (x, y) P (y, z) P (x, z))) which express that the binary predicate P is re exive, symmetric and transitive, respectively. Show that none of these sentences is semantically entailed by the other ones by choosing for each pair of sentences above a model which satis es these two, but not the third sentence essentially, you are asked to nd three binary relations, each satisfying just two of these properties..

2 Predicate logic 7. Show the barcode 3 of 9 for None semantic entailment x x ; for that you have to take any model which satis es x and you have to reason why this model must also satisfy x . You should do this in a similar way to the examples in Section 2.

4.2. * 8.

Show the semantic entailment x P (x) x Q(x) x (P (x) Q(x)). 9. Let and and be sentences of predicate logic.

(a) If is semantically entailed by , is it necessarily the case that is not semantically entailed by * (b) If is semantically entailed by , is it necessarily the case that is semantically entailed by and semantically entailed by (c) If is semantically entailed by or by , is it necessarily the case that is semantically entailed by (d) Explain why is semantically entailed by i is valid. 10. Is x (P (x) Q(x)) x P (x) x Q(x) a semantic entailment Justify your answer.

11. For each set of formulas below show that they are consistent: (a) x S(x, x), x P (x), x y S(x, y), x (P (x) y S(y, x)) * (b) x S(x, x), x y S(x, y), x y z ((S(x, y) S(y, z)) S(x, z)) (c) ( x (P (x) Q(x))) y R(y), x (R(x) Q(x)), y ( Q(y) P (y)) * (d) x S(x, x), x y (S(x, y) (x = y)). 12.

For each of the formulas of predicate logic below, either nd a model which does not satisfy it, or prove it is valid: (a) ( x y (S(x, y) S(y, x))) ( x S(x, x)) * (b) y (( x P (x)) P (y)) (c) ( x (P (x) y Q(y))) ( x y (P (x) Q(y))) (d) ( x y (P (x) Q(y))) ( x (P (x) y Q(y))) (e) x y (S(x, y) ( z (S(x, z) S(z, y)))) (f) ( x y (S(x, y) (x = y))) ( z S(z, z)) * (g) ( x y (S(x, y) ((S(x, y) S(y, x)) (x = y)))) ( z w (S(z, w))). (h) x y ((P (x) P (y)) (P (y) P (x))) (i) ( x ((P (x) Q(x)) (Q(x) P (x)))) (( x P (x)) ( x Q(x))) (j) (( x P (x)) ( x Q(x))) ( x ((P (x) Q(x)) (Q(x) P (x)))) (k) Di cult: ( x y (P (x) Q(y))) ( y x (P (x) Q(y)))..

Exercises 2.5 1. Assuming Software barcode 3 of 9 that our proof calculus for predicate logic is sound (see exercise 3 below), show that the validity of the following sequents cannot be proved by nding for each sequent a model such that all formulas to the left of evaluate to T and the sole formula to the right of evaluates to F (explain why this guarantees the non-existence of a proof):. 2.8 Exercises (a) x (P (x Software USS Code 39 ) Q(x)) x P (x) x Q(x) * (b) x (P (x) R(x)), x (Q(x) R(x)) x (P (x) Q(x)) (c) ( x P (x)) L x (P (x) L), where L has arity 0 * (d) x y S(x, y) y x S(x, y) (e) x P (x), y Q(y) z (P (z) Q(z)). * (f) x ( P (x) Q(x)) x (P (x) Q(x)) * (g) x ( P (x) Q(x)) x (P (x) Q(x)). 2.

Assuming that is sound and complete for in rst-order logic, explain in detail why the undecidability of implies that satis ability, validity, and provability are all undecidable for that logic. 3. To show the soundness of our natural deduction rules for predicate logic, it intuitively su ces to show that the conclusion of a proof rule is true provided that all its premises are true.

What additional complication arises due to the presence of variables and quanti ers Can you precisely formalise the necessary induction hypothesis for proving soundness .
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