Part I: Understanding logical expressions (30 pt.) (10 pt., 2 pt. each) Write ea

Question

Part I: Understanding logical expressions (30 pt.) (10 pt., 2 pt. each) Write each of the following conditional statements in the form "if p, then q" and state the corresponding converse, inverse, and contrapositive a. I will send you the address only if you give me the number b. Getting the job implies that you were the best candidate. c. It is necessary to have a valid password to log on to the server. d. A sufficient condition for the warranty to be good is that you bought the computer less 1. than a year ago. e. You will pass the course unless you do not complete the homework. 2. (10 pt., 2 pt. each) Let P(x) be the statement "x is a student" (x) be the statement "x is a professor," and R(x, y) be the statement "x asked a question to y." If the domain of x and y consists of all people, express each of the following sentences in terms of P(x), Q(x), R(x, y), quantifiers, and logical operators. n a. It is not the case that some student has never asked a question.ar ons yb. There is a professor who has never been ásked a question by a student. c. Some student has asked every professor a question. d. There is a student who has asked a question to exactly one professor e. There are two different students who have asked each other a question. 3. (10 pt, 2 pt. each) Determine the truth value of each of the following statements if the domain consists of all integers. If it is true, find a domain for which it is false. If it is false, find a domain for which it is true. Justify your answer.

Explanation / Answer

converse-If q , then p

Inverse-If not p , then notq .

If you bought the computer not less than one year ago, then the sufficient condition for warranty is not good

Part 2:

Let

P(x) be the predicate "x is a student"

Q(x) be the predicate "x is a faculty member"

a.) ¬R(x,y) //since negation of a student asking a faculty

b.)   x(P(x) y(Q(y) ¬R(y, x)))

c.) y(P(y) x(Q(x) R(x, y)))

d.)   R(x,y) //since a student asking a faculty

e.) R(x,y) <->R(y,x)

Qstn No statement- If p , then q

converse-If q , then p

Inverse-If not p , then notq .

Contrapositive-If not q , then notp . a I will send you the address only when you give me the number if you give me the number,then I will send you the address If I will not send you the address, then you donot give me the number If you donot give me the number, then I will not send you the address. b Getting the job implies that you were the best candidate If you were the best then you are getting the job If you are not getting the job then you were not the best candidate If you were not the best candidae then you are not getting the job c It is necessaryto have a valid password to log on to the server. If there is a need to log on to the server then it is necessary to have a valid password. if It is not necessary to have a valid password then cannot log on to the server If not logged on to the server then it is not necessary to have a valid password d A sufficient condition for the warranty to be good is that you bought the computer less than one year ago. If you bought the computer less than one year ago, then the sufficient condition for warranty is good. If the sufficient condition for warranty is not good, then you bought the computer not less than one year ago

If you bought the computer not less than one year ago, then the sufficient condition for warranty is not good

e You will pass the course unless you donot complete the homework If you complete the homework then you will pass the course if you will not pass the course then you donot complete the homework If you do not complete the homework, then you will not pass the course

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