I encountered a proof that the empty set is a subset of every set via this comment(Is "The empty set is a subset of any set" a convention?) which shows that it cannot be false that the empty set is a subset of every set. Without necessarily going into a proof of how the empty set is a subset of every set, I was wondering why the fact that it cannot be false that the empty set is a subset of every set shows that this is true- could it not be the case that the concept of subsets is meaningless with regards to the empty set, and it is not enough to show that it could not be false; that this statement could neither true or false as it has no meaning in this context?
Also, I would appreciate some explanation as to how this condition holds "vacuously" as far as terminology, as I have learned that for an implication to be vacuously true, it is true when it's hypothesis is false.
Thanks
Best Answer
You can prove that the empty set is a subset of every set by going to the definition. In one sense that says it is not a convention, but the edge cases of definitions are chosen to make things as clean as possible. One could claim that it is a convention that subset is defined this way. In this case it seems very natural to have the definition so that it is true.
The definition of $A \subset B$ is that all elements of $A$ are also elements of $B$. For the empty set we say it is vacuously true because there are no elements of $\emptyset$. If we expand the abbreviation of subset we get $\forall x (x\in A \implies x \in B)$ As there are no $x \in \emptyset$ when $A=\emptyset$ the antecedent is always false so the implication is always true. That is what we mean by vacuous truth.