Show that the two quantified statements in each problem are not logically equivalent by filling in a table so that, for the domain (a, b, c), the values of the predicate P you select for the table causes one of the statements to be true and the other to be false. For example, the table below shows that vx vy P(x, y) and 3x3y P(x, y) are not logically equivalent because for the given values of the predicate P, vx vy P(x, y) is false and 3x 3y P(x, y) is true. P a b C a T T T b T F T T T F (a) vx³y P(x, y) and 3x vy P(x, y) (b) vx³y ((x+y) P(x, y)) and vx 3y P(x, y) (c) 3x³ (P(x, y) AP(y, x)) and 3x3 P(x, y)

Express this in predicate logic: Each email address has exactly one email box. M(x): x is an email address B(x,y): x has an email box y

Apply De Morgan's law to the following expression to obtain an equivalent expression in which each negation sign applies directly to a predicate. (P(x) ^ (Q(x) v R(x))) O 3x (-P(x) v (¬Q(x) ^ ¬R(x))) -3x P(x) or 3x ¬(P(x) ^ Q(x)) O 3x (P(x) v (Q(x) ^ ¬R(x))) O None are equivalent.

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