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Let F(x,y) be the statement “x can fool y,” where the domain consists of all people in the world.Use quantifiers to express each of these statements. (1.5) a) Everybody can fool Fred. b) Evelyn can fool everybody. c) Everybody can fool somebody. d) There is no one who can fool everybody. e) Everyone can be fooled by somebody. f) No one can fool both Fred and Jerry. g) Nancy can fool exactly two people.

Let C(x) be the statement “x has a cat,” let D(x) be the statement“x hasadog,”and let F(x) bethestatement“x hasaferret.”Express each of these statements in terms of C(x), D(x), F(x), quantifiers, and logical connectives. Let the domain consist of all students in your class. (1.4) a) A student in your class has a cat, a dog, and a ferret. b) Allstudentsinyourclasshaveacat,adog,oraferret. c) Some student in your class has a cat and a ferret, but not a dog. d) Nostudentinyourclasshasacat,adog,andaferret. e) For each of the three animals, cats, dogs, and ferrets, thereisastudentinyourclasswhohasthisanimalas a pet.

Show that (p → q) ∨ ( p → r) and p → (q ∨ r)are logically equivalent.(1.3)(no truth table) Show that each of these conditional statements is a tautology by using truth tables d) [ (p ∨ q)∧(p → r) ∧ (q →r)] → r

Let p and q be the propositions p :I bought a lottery ticket this week. q :I won the million dollar jackpot. Express each of these propositions as an English sentence. (1.1) a) ¬p b) p∨q c) p →q d) p∧q e) p ↔q f) ¬p →¬q g) ¬p∧¬q h) ¬p∨(p∧ q)

Write each of these statements in the form “if p, then q” inEnglish.[Hint:Refer to the list of common ways to Express conditional statements provided in this section.] (1.1) a) I will remember to send you the address only if you send me an e-mail message.

b) To be a citizen of this country, it is sufficient that you were born in the United States. c) If you keep your text book,it will beause ful reference in your future courses. d) The Red Wings will win the Stanley Cup if their goalie plays well. e) That you get the job implies that you had the best credentials. f) The beach erodes whenever there is a storm. g) It is necessary to have a valid password to log on to the server.

h) Youwillreachthesummitunlessyoubeginyourclimb too late.

Construct a truth table for each of these compound propositions(1.1) d) (p∨q)→(p∧q) e) (p →q)↔(¬q →¬p)

For each of these sets of premises, what relevant conclusion or conclusions can be drawn ? Explain the rules of inference used to obtain each conclusion from the premises.(1.6) d)“Every student has an Internet account.”“Homer does not have an Internet account.”“Maggie has an Internet account.”

e) “All foods that are healthy to eat do not taste good.” “Tofu is healthy to eat.” “You only eat what tastes good.” “You do not eat tofu.” “Cheeseburgers are not healthy to eat.”

Use a direct proof to show that the product of two odd numbers is odd.(1.7)

Prove that if n is an integer and 3n+2 is even, then n is even using

a) a proof by contraposition.

b) a proof by contradiction.(1.7)...