# Math Help - Explaining the empty set: subset or element of set

1. ## Explaining the empty set: subset or element of set

Greetings, I have a question regarding the difference between an element of a set and a subset, with regards to the empty set. What is the difference between the two statements:

(a) Is x an element of {x} and (b) Is x a subset of {x} ?

I understand that the empty set is a subset of every set, but I don't see why it, as a subset, is not considered an element of every set.

2. ## Re: Explaining the empty set: subset or element of set

Originally Posted by aprilrocks92
Greetings, I have a question regarding the difference between an element of a set and a subset, with regards to the empty set. What is the difference between the two statements:
(a) Is x an element of {x} and (b) Is x a subset of {x} ?
I understand that the empty set is a subset of every set, but I don't see why it, as a subset, is not considered an element of every set.
The statement that "the empty set is an element of every set" is FALSE. It is not.

Let $S=\{x,y,z\}$ then $\emptyset\notin S~.$ It is not listed there.

Let $T=\{x,y,\emptyset\}$ then $\emptyset\in T~.$ It is listed there.

3. ## Re: Explaining the empty set: subset or element of set

Rudin says the empty set is a subset of every set because you can't show that it is not a subset. I know, tricky.

OK. Plato is right. A subset of a set is not a member of a set unless specifically included in the set. {a,b} is a subset of {a,b,c} but not a member. {a,b} is a subset of {a,b,{a,b},c} and a member.

Sorry, repititious, just trying to fix it in my mind.

4. ## Re: Explaining the empty set: subset or element of set

Originally Posted by aprilrocks92
I understand that the empty set is a subset of every set, but I don't see why it, as a subset, is not considered an element of every set.
The contents of the empty set (nothing) is an element of every set. I don't know if there is a symbol for the contents of the empty set.

5. ## Re: Explaining the empty set: subset or element of set

Originally Posted by aprilrocks92
but I don't see why it, as a subset, is not considered an element of every set.
You seem to be under the impression that a "subset" is an "element". That is not true. An "element of a set" is one of the members of the set. A subset is a set of objects that happen to also be in the original set. {x} contains the element x. x, here, is not a "subset" of {x} because it is NOT a set!

6. ## Re: Explaining the empty set: subset or element of set

Originally Posted by Hartlw
The empty set is a subset of every set. I know, tricky.
It is not tricky at all:
$(\forall x~\&~\forall A)[\text{ if }x\in\emptyset\text{ then }x\in A\text{ is a true statement.}]$ A false statement implies any statement.
That is the very definition of subset so $\emptyset\subseteq A$.

7. ## Re: Explaining the empty set: subset or element of set

Originally Posted by HallsofIvy
(aprilrocks92) seems to be under the impression that a "subset" is an "element". That is not true. An "element of a set" is one of the members of the set.
Actually, I assumed that aprilrocks92 may have been studying transitive sets.
Much like the enlarged universe of non-standard analysis.

Let $U=A\cup \mathcal{P}(A)$. Now for any set $A$ we have $\emptyset\in U~\&~\emptyset\subseteq U$.

8. ## Re: Explaining the empty set: subset or element of set

Originally Posted by Plato
It is not tricky at all:
$(\forall x~\&~\forall A)[\text{ if }x\in\emptyset\text{ then }x\in A\text{ is a true statement.}]$ A false statement implies any statement.
That is the very definition of subset so $\emptyset\subseteq A$.
You quoted me out of context. That is not what I meant by tricky.

The tricky part is the proof, which is basically: “the assumption that ɸ is not a subset is false, therefore ɸ must be a subset.” The problem with that standard proof is you haven’t shown that ɸ exists in the first place.

9. ## Re: Explaining the empty set: subset or element of set

Originally Posted by Hartlw
the assumption that ɸ is not a subset is false, therefore ɸ must be a subset.” The problem with that standard proof is you haven’t shown that ɸ exists in the first place.
You are a person after R L Moore's own heart.
He has been called the greatest math teacher ever by Keith Devlin.

Moore was one of the founding fathers of Point Set Topology. His foundational book has no empty point set in it. He was stead fast in his denial it was possible. Look at the list of his students. You see there some of the most important mathematicians of the 20th century. On that list is Mary Ellen Rudin who was married to your offed quoted Walter Rudin. Moore is the only person who has had five of his students to be president of the MAA.

Of course, all of mathematics is the result of the human brain. It is from definitions and/or axioms. Existence is not an issue. The empty set is very much part of modern mathematics.

10. ## Empty Set

Set: A collection of objects.
Empty Set: A set with no objects.
Therefore:
The empty set does not exist.
Therefore:
Rudin’s and other’s proof that the empty set is a subset of every set is invalid.

By the way, the universal set U does not exist. As soon as you define it, it becomes a subset of itself, which is a circular, invalid, definition.

The same argument can be used to show that the empty set is not a subset of every set. If it were, it would be a subset of itself and hence undefined.

Vague, abstract, symbology notwithstanding.

11. ## Re: Empty Set

Originally Posted by Hartlw
By the way, the universal set U does not exist. As soon as you define it, it becomes a subset of itself, which is a circular, invalid, definition.

The same argument can be used to show that the empty set is not a subset of every set. If it were, it would be a subset of itself and hence undefined.
Mistake in first sentence. It should be:
By the way, the universal set U does not exist. As soon as you define it, it becomes a member of itself, which is a circular, invalid, definition.

The second sentence is wrong because any set is a subset of itself. There is no inconsistency.

Basically, if a set is defined as a collection of objects, the empty set doesn't exist and you can't draw conclusions about it based on set-theoretic arguments, like "the empty set is a subset of every set."

"Empty set" may be useful as a symbol 0' for the phrase "doesn't exist." For example, if A and B have no elements in common, A^B=0' simply says the intersection doesn't exist, rather than the intersection is a new set which doesn't have any members.

If a set consists of a description AND a collection of objects, that's a whole different ball game.

12. ## Re: Explaining the empty set: subset or element of set

Originally Posted by Plato
You are a person after R L Moore's own heart.
He has been called the greatest math teacher ever by Keith Devlin.

Moore was one of the founding fathers of Point Set Topology. His foundational book has no empty point set in it. He was stead fast in his denial it was possible. Look at the list of his students. You see there some of the most important mathematicians of the 20th century. On that list is Mary Ellen Rudin who was married to your offed quoted Walter Rudin. Moore is the only person who has had five of his students to be president of the MAA.

Of course, all of mathematics is the result of the human brain. It is from definitions and/or axioms. Existence is not an issue. The empty set is very much part of modern mathematics.
Thanks for that post. I really didn't appreciate it until exercising the subject:
Set, Empty Set

Basically,
Remove a,b,c from the collection of objects a,b,c and you are left with nothing.
Remove a,b,c from the set {a,b,c} and you are left with {}.