# Are null sets the only ones that are disjoint but not distinct?

While studying analysis (which includes a chapter about set theory) I read this:

Two sets $$A,B$$ are said to be disjoint if $$A ∩ B = ∅$$. Note that this is not the same concept as being distinct, $$A \neq B$$. For instance, the sets $$\{1, 2, 3\}$$ and $$\{2, 3, 4\}$$ are distinct (there are elements of one set which are not elements of the other) but not disjoint (because their intersection is non-empty). Meanwhile, the sets $$∅$$ and $$∅$$ are disjoint but not distinct.

Note: Not including curly braces for the empty sets isn't a typo on my part, I copied everything as it is.

While I understand why these $$2$$ sets are disjoint but not distinct, it seems very illogical to me that any sets with non-null elements would have this property.

I think the same would apply to sets such as $$\{∅,∅\}$$ and $$\{∅,∅\}$$ (someone correct me if I am wrong) but I can't think of any examples such that the set has at least one element that it is not null, so is it the case that null sets the only ones that are disjoint but not distinct (as this wasn't mentioned in the textbook) or am I missing something?

• If $A$ and $B$ are not distinct then $A=B$. If they are also disjoint this implies $A=A\cap A=A\cap B=\emptyset$. May 26, 2020 at 5:29

I think you are a bit confused about set notation. There is a difference between the set $$\emptyset$$ and the set $$\{\emptyset\}$$. The first one is a set with no elements, and it has cardinality $$0$$. The second one is a set with one element, namely the empty set, and it has cardinality $$1$$.
Moreover, a set is a collection of distinct objects. Sets cannot include the same element twice. That is to say, the set $$\{\emptyset, \emptyset\}$$ is non other than just $$\{\emptyset\}$$. It has cardinality $$1$$.
Now, regarding your question, as noted in the comments, if two sets $$A$$ and $$B$$ are disjoint but not distinct then they both have to be the empty set $$\emptyset$$.
Be careful, the sets $$\{\emptyset\}$$ and $$\{\emptyset\}$$ are neither distinct, nor disjoint. We have that $$\{\emptyset\}\cap\{\emptyset\} = \{\emptyset\} \neq \emptyset$$ This might sound a bit confusing at first, but it is absolutely necessary to take the time to understand all of the above.