Using quantifiers to express this sentence.

Question

These are from a study guide, just checking my work.

Let $F(x,y)$ be the statement "$x$ and $y$ are friends." where the domains consists of all people in the class. Use quantifiers to express the following:

A.) "There exists some student in this class who is friends with all of the others."

$\exists x \forall y F(x,y) $

B.) "All students in this class must have at least one friend."

$\forall x \exists y F(x,y)$

C.) "There exists some students in this class whose friends are not friends with each other."

$\exists x \exists y \neg F(x,y)$

Answer 1

Your answers to (a) and (b) are correct.

Your answer to (c) says: there exist two people in the class who are not friends. This is not the same as what the original statement means; but you're very close to having the right answer.

Break it down:

• There exists a student, say $x$...
• ...and given any of this student's friends, say $y$ and $z$, i.e. $F(x,y)$ and $F(x,z)$...
• ...these friends are not friends with each other, i.e. $\neg F(y,z)$.

Can you see how to put this together?

Answer 2

The first two are correct, the third is wrong.

The third sentence says that there is someone, and every two friends that someone has are not friends with one another. This means that there is some student, that for every two distinct students which are his friends, these two are not friends themselves.

(Note that the I require that the two friends are distinct, but this is not necessary if you decree that no one is friends with themselves.)

Answer 3

I think there is a mistake at the last sentence. What you have wrote is interpreted as "There are two people that are not friends". But what you need is a person that will have at least two friends which won't be friends with each other. So actually, you want $\exists x\exists y\exists z(F(x,y)\wedge F(x,z)\wedge(\neg F(y,z)))$.