Exercise 6: (5 POINTS) Suppose the variable 𝑥 represents students and 𝑦 represents courses,and:𝐴(𝑥): 𝑥 is a part-time student𝑀(𝑦): 𝑦 is a math course𝐹(𝑥): 𝑥 is a freshman𝐵(𝑥): 𝑥 is a full-time student𝑇(𝑥, 𝑦): student 𝑥 is taking course 𝑦.1. Write the following statements using these predicates and any needed quantifiers.a) Caroline is not taking any course. (1 POINT)b) No student is taking every course. (1 POINT)2. Write the following statements in good English without using variables in your answers.a) ∃𝑦∀𝑥[𝐴(𝑥) → 𝑇(𝑥, 𝑦)] (1 POINT)b) ∀𝑥∃𝑦[(𝐵(𝑥) ∧ 𝐹(𝑥)) → (𝑀(𝑦) ∧ 𝑇(𝑥, 𝑦))] (2 POINTS)

Let 𝐼(𝑥) be the statement “𝑥 has an Internet connection” and 𝐶(𝑥, 𝑦) be the statement “𝑥 and 𝑦 havechatted over the Internet,” where the domain for the variables 𝑥 and 𝑦 consists of all students in yourclass. Use quantifiers to express each of these statements.a) Jerry does not have an Internet connection.b) Rachel has not chatted over the Internet with Chelsea.c) No one in the class has chatted with Bob.d) Sanjay has chatted with everyone except Joseph.e) Someone in your class does not have an Internet connection.f) Not everyone in your class has an Internet connection.g) Exactly one student in your class has an Internet connection.h) Everyone in your class with an Internet connection has chatted over the Internet with at leastone other student in your class.i) Someone in your class has an Internet connection but has not chatted with anyone else in yourclass.j) There are two students in your class who have not chatted with each other over the Internet.k) There is a student in your class who has chatted with everyone in your class over the Interne

Let 𝐶(𝑥, 𝑦) mean that student 𝑥 is enrolled in class 𝑦, where the domain for 𝑥 consists of all students inyour school and the domain for 𝑦 consists of all classes being given at your school. Express each of thesestatements by a simple English sentence.a) 𝐶(𝑅𝑎𝑛𝑑𝑦 𝐺𝑜𝑙𝑑𝑏𝑒𝑟𝑔, 𝐶𝑆 252)b) ∃𝑥𝐶(𝑥, 𝑀𝑎𝑡ℎ 695)c) ∃𝑦𝐶(𝐶𝑎𝑟𝑜𝑙 𝑆𝑖𝑡𝑒𝑎, 𝑦)d) ∃𝑥(𝐶(𝑥, 𝑀𝑎𝑡ℎ 222) ∧ 𝐶(𝑥, 𝐶𝑆 252))e) ∃𝑥∃𝑦∀𝑧((𝑥 ≠ 𝑦) ∧ (𝐶(𝑥, 𝑧) → 𝐶(𝑦, 𝑧)))f) ∃𝑥∃𝑦∀𝑧((𝑥 ≠ 𝑦) ∧ (𝐶(𝑥, 𝑧) ↔ 𝐶(𝑦, 𝑧)))

Exercise 7: (6 POINTS) Express each of these statements using predicates with two variables andnested quantifiers. Then form the negation of the statement so that no negation is to the left ofa quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It isnot the case that.”)a) Every Ashesi student has taken some math course.b) No one knows everybody.

Let 𝑃(𝑥, 𝑦) be the statement “𝑥 enjoys playing 𝑦,” where the domain for𝑥 consists of all Ashesi students and that of 𝑦 consists of all sports.Express each of these statements by a simple English sentence.a) 𝑃(Paul, football)b) ∃𝑦𝑃(Carol, 𝑦)c) ∃𝑥(𝑃(𝑥, basketball) ∧ 𝑃(𝑥, badminton))d) ∃𝑥∀𝑦((𝑥 ≠Ben) ∧ (𝑃(Ben, 𝑦) → 𝑃(𝑥, 𝑦))

Translate each of the English sentences (a)-(d) below into predicate logic formulas, with all students as the domainof discourse.The only predicates you may use are• O(x) meaning that “x is a computer science student”, and• H(x) meaning that “x is studying hard”, and• M (x, y) meaning that “x has won more Hackathons than y”, and• equality and inequality. (i.e. = and̸ =)However, you may define whatever variables and constants you wish, and you are also allowed to use all the symbolsfrom the alphabet of predicate logic, i.e. connectives, quantifiers, brackets etc.If you need to make any assumptions, state what they are.(a) Some hard-studying students are not computer science students.(b) All computer science students study hard.(c) Some students have won more Hackathons than everyone else.(d) Duana is the only student who has won more Hackathons than Phill