Express the statements using quantifiers example

Question

I'm having a little trouble understanding quantifiers and therefore doubting all my homework answers. Since there is no where to check if the answers are correct, I'm very very worried I am just practicing incorrectly. So I've set up two examples with what I think the answers are.

It would be brilliant if you could confirm if I am correct or not so I could use these answers as a base to check my other answers. If I am incorrect, it would be awesome if you could point me the right direction!

Express the statements using quantifiers. *note: ~ = negation.

a) Everybody knows everybody.

my answer: ∀xE(x), E(x) = knows everybody.

b) Somebody knows everybody.

my answer: ∃xE(x)

c) There is somebody whom no one knows.

my answer: ~∃xE(x)

Answer 1

Rather than using $E(x)$ to mean "$x$ knows everybody", it might be more helpful to use something like $K(x,y)$ to mean "$x$ knows $y$".

Using this, these could be stated as:

a) $\forall x\, \forall y\, K(x,y)$

b) $\exists x\, \forall y\, K(x,y)$

c) $\exists x\, \forall y\, \neg K(y,x)$.

Answer 2

It is probably more in the intent of the exercises to let $R(x, y)$ be "$x$ knows $y$," and to use double-quantifiers... Also, you should technically have a domain (e.g. all people)

But, your answers for a and b are correct (c is wrong, see comments), given your definition of $E$ and assuming you aren't required to list the domain.

For an example of double quantifiers: Let $R(x, y)$ be "$x$ knows $y$," and let $D$ be the set of all people. Then: $$\text{"Everybody knows somebody"} \iff \forall x \in D,\exists y\in D \text{ such that } R(x, y)$$