Sets and subsets: What is the difference between these two statements?

Would it be correct to say that $$\emptyset \subseteq \emptyset$$ or $$\emptyset \subseteq \{\emptyset\}$$? To my understanding, the null set is just an empty set, so a null set is a subset of a set that contains the null set as an element, hence $$\emptyset \subseteq \{\emptyset\}$$ would be true. But wouldn't the first statement also be true?

• Every set is a subset of itself. Empty set is a subset of every set. – user655800 Oct 2 '19 at 4:16

Every set is a subset of itself. And the empty set is a subset of every set. So both statements are true.

However, I suspect that this only adds a bit to the confusion.

Note that:

1. $$\emptyset\subseteq \emptyset$$ (because any set is always a subset of itself), but $$\emptyset\notin \emptyset$$ (because the empty set does not contain any elements, and in particular the empty set is not an element in $$\emptyset$$.

2. $$\emptyset\subseteq\{\emptyset\}$$ (because the empty set is always a subset of any set), but in addition, $$\emptyset\in\{\emptyset\}$$, because $$\emptyset$$ is in fact an element of $$\{\emptyset\}$$, the set whose only element is $$\emptyset$$. In general, any set $$A$$ is an element of the set $$\{A\}$$.

So the following four statements are true:

1. $$\emptyset\subseteq \emptyset$$.
2. $$\emptyset\subseteq \{\emptyset\}$$.
3. $$\emptyset\notin \emptyset$$.
4. $$\emptyset\in\{\emptyset\}$$.
• In regards to point 2, would it also be true that a set A is a subset of {A}? – Random Student Oct 2 '19 at 4:26
• @RandomStudent: No, in general it will not. That would require that if $x\in A$, then $x\in\{A\}$, hence $x\in A$ would imply $x=A$. That would that either there are no such $x$ (so $A=\varnothing$) or else $A$ is not empty and therefore the condition means that $A=\{A\}$. In the usual set theory, no set satisfies that condition. So the only set for which that would be true is $A=\emptyset$. – Arturo Magidin Oct 2 '19 at 4:37

They are both true but mean different things altogether.

$$\emptyset \subset \emptyset$$ is true.

It is true for any of the following reasons and maybe more.

1) Every set is a subset of itself.

2) The emptyset is a subset of any set

3) The emptyset has no elements so every element is in the emptyset is vacuuously in the emptyset

4) there are no elements in the emptyset that are not in the emptyset.

$$\emptyset \subset \{\emptyset\}$$ is also true but it means something entirely different.

It is true because

1) The emptyset is a subset of any set.

2) The emptyset has no elements at all so every element it has is vacuously an element of $$\{\emptyset\}$$.

3) The emptyset doesn't have any elements not in $$\{\emptyset\}$$.

etc.

However the reason you gave:

"so a null set is a subset of a set that contains the null set as an element, hence ∅⊆{∅} would be true."

is completely wrong.

A set, $$A$$ is NOT as subset of a set containing it.

$$A\not \subset \{A\}=B$$

$$A$$ is an element of $$B$$ but the elements within $$A$$ are not elements of $$B$$ at all. We never consider the elements within elements when determining subsets.

That $$A\in B$$ is no more relevant and has no more bearing on whether $$A$$ is subset of $$B$$ then whether $$froofroothedog \in B$$.