I need help understanding the concept of "at most one " using quantifers

Solution 1:

• $$∀x∀y\;\big((\text{Cube}(x) ∧ \text{Cube}(y)) → x = y\big)$$

  Yes the above statement allows for the possibility of exactly one cube in the box, as you have explained.

  But it also allows for the possibility of no cube in the box: in this case, whichever pair of objects you pick, $(\text{Cube}(x) ∧ \text{Cube}(y))$ is false and the entire statement consequently vacuously true.

  Here's an equivalent, more succincct formulation of at most one: $$\exists x\forall p\;(\text{Cube}(p)\to p=x).$$

• In contrast, exactly one cube can be translated as $$\exists x\forall p\;(\text{Cube}(p)\leftrightarrow p=x).$$