Let l(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.

The question is:

Let l(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.

Step-by-step explanation:

Note that: Because the arguments are variables, we write

Ix = I(x)

Cxy = C(x,y)

a. Jerry does not have an Internet connection.

~I(Jerry).

b. Rachel has not chatted over the internet with Chelsea.

~C(Rachel,Chelsea).

c. Jan and Sharon have never chatted over the internet.

~C(Jan,Sharon).

d. No one in the class has chatted with Bob.

~∃x C(x, Bob).

e. Sanjay has chatted with everyone except Joseph.

∀y (y ≠ Joseph → C(Sanjay, y)).

f. Someone in your class does not have an internet connection.

∃x ~Ix.

g. Not everyone in your class has an internet connection.

~∀x Ix.

h. Exactly one student in your class has an internet connection.

∃x (Ix ∧ ∀y (Iy → y = x)).

i. Everyone except one student in your class has an internet connection. This means exactly one student does not have a connection.

∃x (~Ix∧ ∀y (Iy → y = x)).

j. Everyone in your class with an internet connection has chatted over the internet with at least one other student in your class.

∀x (Ix → ∃y (Cxy ∧ x ≠ y)).

Assuming nobody can chat with him or herself, then

∀x (Ix → ∃y (Cxy))

k. Someone in your class has an internet connection but has not chatted with anyone else in your class.

∃x (Ix ∧ ∀y ~Cxy).

l. There are two students in your class who have not chatted with each other over the internet.

∃x ∃y (x ≠ y∧~Cxy).

m. There is a student in your class who has chatted with everyone in your class over the internet.

∃x ∀y Cxy.

n. There are at least two students in your class who have not chatted with the same person in your class. ∃x ∃y (x ≠ y ∧ ∃z (~Cxz ∧ ~Cyz)).

o. There are two students in the class who between them have chatted with everyone else in the class.

∃x ∃y (x ≠ y ∧ ∀z (Cxz ∨ Cyz)).

4x and 6y

im not sure though

example 2 – simplify: step 1: completely factor both the numerators and denominators of all fractions. step 2: change the division sign to a multiplication sign and flip (or reciprocate) the fraction after the division sign; essential you need to multiply by the reciprocal. step 3 : cancel or reduce the fractions.