Question

Let I (x) be the statement "x has an Internet connection" and C(x, y) be the statement "x and y have chatted over the Internet," where the domain for the variables x and y consists of all students in your class. 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) Jan and Sharon have never chatted over the Internet. (d) No one in the class has chatted with Bob. (e) Sanjay has chatted with everyone except Joseph. (f) Someone in your class does not have an Internet connection. (g) Not everyone in your class has an Internet connection. (h) Exactly one student in your class has an Internet connection. (i) Everyone except one student in your class has an Internet connection. (j) Everyone in your class with an Internet connection has chatted over the Internet with at least one other student in your class. (k) Someone in your class has an Internet connection but has not chatted with anyone else in your class. (l) There are two students in your class who have not chatted with each other over the Internet. (m) There is a student in your class who has chatted with everyone in your class over the Internet. (n) There are at least two students in your class who have not chatted with the same person in your class. (o) There are two students in the class who between them have chatted with everyone else in the class

Answer 1

Explanation:

L(x) means that Student X has an internet connection

C(x,y) means that students X and Y have chatted over the internet

The domain for variables X and Y comprise all students in your class. We now use quantifiers or algebraic functions to express each of the statements:

(A) Jerry does not have an internet connection

L(x) = 0

Where X represents Jerry

(B) Rachel has not chatted over the internet with Chelsea

C(x,y) = 0

Where X and Y represent Rachel and Chelsea

(C) Jan and Sharon have never chatted over the internet

L(x) + L(y) = 0

Where X and Y represent Jan and Sharon. If they have NEVER chatted over (the question didn't say they have never "with each other", it says they have never chatted at all) the internet, then they've probably never had an internet connection!

(D) No one in the class has chatted with Bob

Let Y represent Bob and X1, X2, ..., Xn represent everyone else in the class.

The value of Y is not significant here (because it is raised to the power of zero and that makes it equal to 1 and when 1 is multiplied by any X value, the X value or student remains the same) but we have to put it, to represent Bob.

The quantifier here is C (X1Y°, X2Y°, X3Y°, ..., XnY°)