Quantified Statements and Email

[Bashyboy]

Homework Statement

Let M(x,y) be "x has sent y an email message" and T(x,y) be "x has telephoned y," where the domain consists of all students in your class. Use quantifiers to express each of these statements. (Assume that all email messages sent are received, which is not the way things often work.)

a) Chou has never sent an e-mail message to Koko.

b) Arlene has never sent an e-mail message to or telephoned Sarah.

c) José has never received an e-mail message from Deborah.

d) Every student in your class has sent an e-mail message to Ken.

e) No one in your class has telephoned Nina.

f) Everyone in your class has either telephoned Avi or sent him an e-mail message.

g) There is a student in your class who has sent everyone else in your class an e-mail message.

h) There is someone in your class who has either sent an e-mail message or telephoned everyone else in your class.

i) There are two different students in your class who have sent each other e-mail messages.

j) There is a student who has sent himself or herself an e-mail message.

k) There is a student in your class who has not received an e-mail message from anyone else in the class and who has not been called by any other student in the class.

l) Every student in the class has either received an email message or received a telephone call from
another student in the class.

m)There are at least two students in your class such that one student has sent the other e-mail and the second student has telephoned the first student.

n) There are two different students in your class who between them have sent an e-mail message
to or telephoned everyone else in the class.

Homework Equations

The Attempt at a Solution

(b) Why is the answer to part (b) $\neg M(Arlene, Sarah) \wedge \neg T(Arlene, Sarah)$, and not $\neg M(Arlene, Sarah) \vee \neg T(Arlene, Sarah)$?

(k) Why is the answer to part (k) $\exists x \forall y (x \ne y \implies (\neg M(x,y) \wedge \neg T(y,x)$, and not $\exists x \forall y (x \ne y \implies (\neg M(y,x) \wedge \neg T(y,x)$?

 

[verty]

With regards to your first question, have you seen this puzzle?

Q: I have two coins in my pocket. They add up to 60 cents. One of them is not a 50c piece. Which two coins do I have?
A: A 50c and a 10c. The one that isn't a 50c is a 10c, the other is a 50c.

Can you now answer your first question?

Your answer to k) is correct.

 

[Bashyboy]

I am not capable of reconciling the first question with the puzzle you have supplied.

 

[Infrared]

Arlene has never sent an e-mail message to or telephoned Sarah means that Arlene did not send an e-mail to Sarah AND that Arlene did not telephone Sarah. Using or means that she might have done one of these.

 

[Bashyboy]

HS-Scientist, so then why does the answer include the conjunction, rather than the disjunction; after all, the word "or" is generally associated with the disjunction.

 

[Infrared]

HS-Scientist, so then why does the answer include the conjunction, rather than the disjunction; after all, the word "or" is generally associated with the disjunction.

Because $\neg (A \lor B)$ is equivalent to $\neg A \land \neg B$