How to write in logical notation "At least n elements" and "Exactly n elements"? $$
 \exists x \exists y \exists z (x\neq y  \land x \neq z \land y \neq z \land P(x) \land P(y) \land P(z))
$$ 
I would have translated the above expression with: "There exist 3 different elements (x,y,z) to which P(x), P(y), and P(z) is true."
In the above example does it say at least 3 elements or exactly 3 elements? I think it's at least 3. If so, how do I write exactly 3 elements?
 A: $P$ holds for at least three elements:
$$\exists x\exists y\exists z\ [x\ne y\land x\ne z\land y\ne z\land P(x)\land P(y)\land P(z)]$$
$P$ holds for at most three elements:
$$\forall w\forall x\forall y\forall z\ [P(w)\land P(x)\land P(y)\land P(z)\to (w=x\lor w=y\lor w=z\lor x=y\lor x=z\lor y=z)]$$
$P$ holds for exactly three elements:
$$\exists x\exists y\exists z\forall w\ [x\ne y\land x\ne z\land y\ne z\land (P(w)\leftrightarrow(w=x\lor w=y\lor w=z))]$$
A: It says "at least three elements". Imagine a structure with four elements in which $P$ is true for all of them; clearly the sentence is satisfied in this structure.
A quick way to say "precisely three" is "at least three, and not at least four".
A: Something like
$$\forall a\,\big(P(a)\Rightarrow (a=x)\vee (a=y)\vee (a=z)\big)$$
says that the property $P(a)$ is only satisfied if $a=x,y,z$. 
A: First establish a set B to which $x,y,z$ may belong then
\begin{equation}
\exists A\subset B\,(\,\vert A\vert=3 \land (\wedge_{x\in A}P(x)\,)
\end{equation}
A: Okay, no-ones suggesting this and maybe it isn't valid but...
isn't $\exists ! a$ meaning there exists a unique $a$ acceptable, and....
isn't $\exists \{a,b,c\}$ meaning there exists a set f three elements acceptable?
If so (and maybe those notations aren't acceptable) wouldn't $\exists! \{x,y,z\} ...$ do it?  There is a unique set of three elements where the conditions hold so there are three elements.  If there were 4 or more elements where the conditions hold and we can make multiple different sets of 3 where the conditions hold.  But as any set of three must be unique, there can only be those three elements.
