English to formal predicate Logic Convert the following into formal predicate logic. Define predicates as necessary. Then negate the predicate sentence. Push all negations to the closest terms.
1) There are at least two people who everyone knows. Domain = {People}
2) Every student takes at least two classes. Domain = {people, classes}
3) All Students know each other. Domain ={ All people} 
My take in this....
for the 1) part is it valid to do something like this ∃∃∀z(, ,z). Where in my words i could be totally wrong.. there exist a pair (x,y) who everyone (z) knows. 
For the 2) part i know that there has to be and existential quantifier and a universal. but I'm not sure how to write them down. 
For the Last part 3) is it this ∀∀(, ) just on the basis that for all X there is a Y 
 A: 

1) There are at least two people who everyone knows. Domain = {People}

My take in this.... for the 1) part is it valid to do something like this ∃∃∀z(, ,z). Where in my words i could be totally wrong.. there exist a pair (x,y) who everyone (z) knows. 

You must say: "There are some $x$ and some $y$ who are not the same people and every $z$ will know $x$ and know $y$."
You should also use a bivariate predicate such as $\def\op#1{\operatorname{\rm #1}}\op{K}(~,~)$ for "_ knows _" 
$$\exists x~\exists y~\forall z~\bigl(x\neq y\wedge \op{K}(z,x)\wedge \op{K}(z,y)\bigr)$$


2) Every student takes at least two classes. Domain = {people, classes}

"For every $x$ who is a student, then there is an $y$ which is a class that is taken by $x$ and there is a $z$ which is another class that is taken by $x$." 
You will need predicates for: $\op P(~)$ "_ is a people", $\op C(~)$ "_ is a class", and $\op T(~,~)$ "_ takes _" .


3) All Students know each other. Domain ={ All people}

"For every $x$ who is a student, then for every $y$ who is a student, then $x$ knows $y$."  
Use predicates $\op S(~)$ for "_ is a student", and $\op{K}(~,~)$ for "_ knows _".
(NB: do you need to worry about whether $x$ and $y$ are the same people?)
