The question is:
Minesweeper is a single-player computer game invented by Robert Donner in 1989. A unary predicate mine is defined, where $mine(x)$ means that the cell $x$ contains a mine
Consider the statement: $$\exists x_1, \dots , \exists x_n \left(\bigwedge_{i=1}^{n} \text{mine}(x_i) \wedge \forall y \left( \text{mine}(y) \to \bigvee_{i=1}^{n} y = x_i\right) \right)$$
Which of the following statements is the correct interpretation of the above formula?
a.There are exactly n mines in the game
b.There are at least n mines in the game
c. There are at most n mines in the game
The way I did this was that the first inner predicate means that are $n$ mines present, implying that there are at least $n$ mines. The second part of the predicate says that if it is a mine, it has be present in one of the $n$ cells, thus implying that there are at most $n$ mines. Hence, the answer should be that there are exactly $n$ mines.
However, the answer given is C i.e there can be at most $n$ mines. One possible reasoning I could think for this is that the second predicate doesn't mean all the $x_i$s have to be different, so there can be less than $n$ mines. However, I don't know why this should be true.
Is my solution correct or is it possible that two of the $x_i$s can refer to the same cell?
The mines are precisely the $x_i$, so there are at most $n$ of them (with exactly $n$ iff the $x_i$ are pairwise distinct, which isn't given).