Examination Sheet № 14
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| Quantifiers. Existential Quantification, Universal quantification. Free and bound variables. Examples. Table of Negating Quantifiers. Prenex Normal Form of Formulae. Show that  x(P(x)  Q(x)) and  x P(x)   x Q(x) are logically equivalent. Show that x(P(x)  Q(x)) and x P(x)  x Q(x) are not logically equivalent. Show that  x(P(x)  Q(x)) and  x P(x)  x Q (x) are logically equivalent. Show that x(P(x)  Q(x)) and x P(x)   x Q (x) are not logically equivalent.
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| r -permutations and r -combinations. Prove theorems on P(n,k), C(n,k).
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| Construct the DNF, CNF and a polynomial for a proposition F (p, q, r) which is true iff (p, q, r) are from {(1, 1, 0), (1, 0, 0), (0, 0, 1), (0, 0, 0)}
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| How many ways to select 23 ordered elements from a set consisting of 54 elements are there, if repetition is not allowed?
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| Find a sequence { an: n = 0, 1, …} satisfying the recurrence relation xn + 6 = –6(xn –1 – 1) – 9 xn –2, with the initial conditions a 0 =1, a 1 = –9.
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2015-07-14
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