There is a student who has e-mailed at most two other people in the class, besides possibly himself.
Let the set of students be $S$, which is the domain of discourse. The only predicates allowed to be used are :
- Equality
- $E(x,y)$, meaning that "x has sent e-mail to y"
Here's my attempt at it, but I cannot be sure how to incorporate the "besides possibly himself" fact into this... The following is what I've done so far:
$$\exists s \in S. (\exists x, \exists y (E(s,x)\land E(s,y))\land (x \neq y))$$
EDIT: My current attempt for this is $$\exists s \in S. (\exists x,y \in S(E(s,x)\land E(s,y) \lor E(s,s)) \land (x \neq y \neq s))$$
Hopefully someone can comment on whether that is accurate or not?
You are saying that there are at least (not at most) two people (one of them might include the student) s.t. the student emailed them. In fact, what you want to express is that if there are three people who have emailed the student that aren't the student, there must be equality between a pair of those three.
So here is how you would say that $$ \begin{align*} &\exists s\in S \text{ s.t.} \forall x,y,z \in S, \\ &(E(s,x) \land E(s,y) \land E(s,z) \land s\neq x \land s\neq y \land s\neq z) \\ &\implies (x=y \lor x=z \lor y=z) \end{align*} $$