Exercise 3: (4 POINTS) Write each of these statements in the form “if 𝑝, then 𝑞” in English.a) He gets sick whenever the weather gets cold. (1 POINT)b) It is necessary to wake up early to beat today’s traffic. (1 POINT)c) A positive integer is a prime only if it has no divisors other than 1 and itself.(1 POINT)d) John will go to Dubai during the mid-semester break unless he runs out of money.(1 POINT

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)

Exercise 2: Put together two parts of these conditional clauses.1 If I had enough money,2 If I have free time,3 If I am in the USA,4 If he is healthy,5 If he got a good mark,6 If he wasn't busy,7 If I stay at home,8 If he were a movie star, Chuck NorrisA I will watch TV.B he could go out.C he would be rich.D he will go to the swimming pool.E I will go to the cinema.F I will visit Chuck Norris.G I would buy a yacht.H he would cook dinner.1..... 2....3.....4....5.....6....7...8​

Write each of these statements in the form “if 𝑝, then 𝑞” in English.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 textbook, it will be a useful 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) You will reach the summit unless you begin your climb too late.

Let 𝑝 and 𝑞 be the proposition variables denoting𝑝: It is below freezing.𝑞: It is snowing.Write the following propositions using variables, 𝑝 and 𝑞, and logical connectives.a. It is below freezing and snowingb. It is below freezing but not snowingc. It is not below freezing and it is not snowing.d. It is either snowing or below freezing (or both).e. If it is below freezing, it is also snowing.f. It is either below freezing or it is snowing, but it is not snowing if itis below freezing.g. That it is below freezing is necessary and sufficient for it to besnowing

Analyze the logical forms of the following statements:(a) 3 is a common divisor of 6, 9, and 15. (Note: You did this in exercise2 of Section 1.1, but you should be able to give a better answer now.)(b) x is divisible by both 2 and 3 but not 4.(c) x and y are natural numbers, and exactly one of them is prime.