Let P(x) be the statement "x can speak Russian" and let Q(x) be the statement "x knows the computer language C++." Express each of these sentences in terms of P(x), Q(x), quantifiers, and logical connectives. The domain for quantifiers consists of all students at your school.

a. Construct an interpretation (i.e., give the domain and the meaning of P(x)) in which (∀x)P(x) has the value true.b. Construct an interpretation in which (∀x)P(x) has the value false.c. Can you find one interpretation in which both (∀x)P(x) is true and (∃x)P(x) is false?d. Can you find one interpretation in which both (∀x)P(x) is false and (∃x)P(x) is true?

Please arrange in a correct order steps of the given argument in logical form. "Everyone in New Jersey lives within 50 miles of the ocean. Someone in New Jersey has never seen the ocean. Therefore, someone who lives within 50 miles of the ocean has never seen the ocean." v (+) 3x(P(x) ^ Q(x)) Existential generalization of (%) v (%) P(c) ^ Q(c) Conjunction from ($), (&) v (&) There is x=c in the domain such that P(c) Universal instantiation (*) - (*) ♥xP(x) Hypothesis Let P(x) = "x lives within 50 miles of the ocean", and Q(x) = "x has never seen the ocean" with the domain of the variable consisting of all people who live in New Jersey. The argument in the logical form: "Everyone in New Jersey lives within 50 miles of the ocean"= VxP(x) "Someone in New Jersey has never seen the ocean" = 3xQ(x) Someone who lives within 50 miles of the ocean has never seen the ocean =3(P(x) ^ Q(x))

2. Use your own predicates P(x) and Q(x) to explain why (3r, P(x)) ^ (3x, Q(x)) is not logically equivalent to 3r, (P(x) A Q(x)).