Empty Set is not a Subset of S

Definition: The empty set E is a subset of S if every member of E is a member of S.

P (Proposition): E is a subset of S.

Proof: P is false if there are members of E which do not belong to S. There are no members of E which do not belong to S, so P is true.

Problem with proof: There are no members of E. So though I can talk about no members, I can’t talk about “no members __which”__ because member doesn’t exist, __which__ doesn’t refer to anything. The problem already occurs in the Definition: to say that “something that doesn’t exist __is__” is meaningless, it says that something that isn’t, is. E may exist, but its members do not.

Conclusion: E is not a subset of S

Re: Empty Set is not a Subset of S

In the case of the empty set, it vacuously belongs to every subset in existence, the empty subset even belongs to the empty subset. It is sort of like the logical proposition $\displaystyle A --> B $ if A is false then the if A then B statement is vacuously true, intuitively it is hard to understand but thats why most definitions state that the empty set is vacuously a subset of every set.

Re: Empty Set is not a Subset of S

intuitively i tend to think of it like this:

Let A,B be any sets. B is a subset of A if $\displaystyle A \cup B = A$.

Now $\displaystyle E \cup B = B$ so the empty set is a subset of every set, B.

You can argue similarly with intersections (ie A is a subset of B if $\displaystyle A \cap B =A$).

Re: Empty Set is not a Subset of S

My conclusion was based on conventional imprecise incorrect definitions of set. A collection of wolves is not a set. An empty collection of wolves is meaningless- my kitchen is an empty collection of wolves?

On the other hand, a pack of wolves makes sense- “pack” is a container consisting of rules which determine membership. There are wolves around which do not satisfy the criteria for membership in the pack. An empty pack means there are no wolves around which satisfy criteria for belonging to the pack. An empty pack is like an empty egg basket.

A precise definition of Set, Subset, and Empty Set is given in my post of the same name in this forum. If I can figure out how to give a direct link to that post I will do so. Looks like this is it:

http://mathhelpforum.com/peer-math-r...empty-set.html