6. Given that P(x, y, z) is a propositional function such that the universe for x, y, z is {1,2}. Furthermore, suppose that the propositional function is true in the following cases, namely: P(1, 1, 1),P(1, 2, 1),P(1, 2, 2),P(2, 1, 1),P(2, 2, 2); and it is false, otherwise. Therefore, determine the truth value of each of the following quantified statements, viz: (a) ∀x∀y∃zP(x, y, z) (b) ∀x∃y∃zP(x, y, z) (c) ∀y∀z∃xP(x, y, z) (d) ∀x∃y∀zP(x, y, z)
6. Given that P(x, y, z) is a propositional function such that the universe for x, y, z is {1,2}.
Furthermore, suppose that the propositional function is true in the following cases, namely:
P(1, 1, 1),P(1, 2, 1),P(1, 2, 2),P(2, 1, 1),P(2, 2, 2); and it is false, otherwise. Therefore, determine
the truth value of each of the following quantified statements, viz:
(a) ∀x∀y∃zP(x, y, z)
(b) ∀x∃y∃zP(x, y, z)
(c) ∀y∀z∃xP(x, y, z)
(d) ∀x∃y∀zP(x, y, z)
7. Let P denote the set of all phones in the world such that p ∈ P is a phone. Thus, S(p)
denotes that “p is a SamsungTM phone”, N(p) denote that “p is a NokiaTM phone”, and G(p)
denote that “p is a GoogleTM phone”. Therefore, express each of the following statements using
quantifiers, logical operations, and the propositional functions: S(p),N(p),G(p).
(a) There is a GoogleTM phone that is also a SamsungTM phone.
(b) Every NokiaTM phone is a GoogleTM phone.
(c) No NokiaTM phone is a GoogleTM phone.
(d) Some NokiaTM phones are also SamsungTM phones.
(e) Some NokiaTM phones are also GoogleTM phones and some are not.
8. Let the universe of discourse be U = {−2,−1, 0, 1, 2}. Compute the negations for each of the
following statements, via transforming the negation connectives inward, such that the negation
symbols immediately precede predicates. Thereafter, determine whether each of the negated
statement is true or false. Note that x and y range over U.
(a) ∀x∃y(x + y = 1).
(b) ∃x∀y(x + y = −y).
(c) ∀x∃y(xy ≥ y).
(d) ∃x∀y(x ≤ y).
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