2) Let S(x,y) be the statement "x is friends with y," where the domains for x and y are all people in the world. Translate each of the following English statements into a formal logical expression using quantifiers. (Each part is worth 5pts.] a) Everyone has a friend. b) Someone is friends with Jamal. c) If you are friends with Mary, then you are also friends with Tim. d) Everyone is friends with themselves. 3) Translate the following formal quantified statements into clear English sentences that convey the same meaning, where the predicate functions have the meaning. Each part worth 5pts.): i. P(w):" computer w is brokeri" ii. Q(w): "computer w is idle" iii. RG); "task j is complete" iv. S():"task is on hold" a) Vw(P(w) VQ(w)) 3/SC) b) VW(-Q(w) -P(w)) c) 3GP(w) V -Q(w) 4) Answer each of the following: a) [5 pts.] Express the negation of this statement so that all negation symbols immediately precede the predicates. 3xVy(Q(x,y)P(xy)) b) [5 pts.] The domains for x and y are all real numbers. Translate the formal quantified statement into English and give its truth value. VxVy((x + 0) (y=0) - (xy # 0)) 5) For each argument below determine whether it is valid or not and explain why using rules of inference or fallacies. a) (10 pts. Consider the following argument: All parrots like fruit. My pet bird is not a parrot. Therefore, my pet bird does not like fruit b) (10 pts. Consider the following argument: No man is an island. Manhattan is an island. Therefore, Manhattan is not a man. 6) (20 pts. Prove that if n is a positive integer, the n is even if and only if 7n + 4 is even. Page 2 of 3 7) (3pts. Extra credit] State the converse, contrapositive and inverse of the following statement "I take my umbrella whenever it rains." 00