Let K be the subset of |R (real numbers:
Statement:
John likes K if and only if ∃a∈K such that ∀x∈K, a < x
Question: Does John like any subset of the real numbers?
My answer: John will not like any subset of the real numbers. This is because regardless of what subset there is, there will never be an element 'a' in the subset K that is less than all the elements in the subset K.
My confusion was: What if the subset was an empty set. Since there are no elements in the empty set, will it make this statement vacuously true? If it does, does that mean John does like the empty set?
If $k=\emptyset$, we have $\exists a \in \emptyset\,\forall x \in \emptyset\,(a<x)$. This is equivalent to $\exists a[a\in \emptyset \land \forall x \in \emptyset\,(a < x)]$. Because $a\in \emptyset$ is a always false, $a\in \emptyset \land \forall x \in \emptyset\,(a < x)$ is always false. Thus $\exists a[a\in \emptyset \land \forall x \in \emptyset\,(a < x)]$ is false, so John does not like the empty set.
Also note that no nonempty subset of $\mathbb{R}$ can have an element less than all elements in that subset because no $x\in\mathbb{R}$ can be less than itself.