Does null set contain one element or no elements?

Question

Let, A is a null set.Now which of the following is true:-

$n(A) = 1$ or,

$n(A) =0$

I knew previously that null set has 0 elements. But I saw in quora that it contains 1 element. So now I'm confused.

If there's any problem in my question please inform me. Thanks!

Answer 1

You were correct the first time.

$\emptyset = \{\}$ is a set that has no elements. So by definition the number of elements it has is .... zero.

However, sets can have sets as elements. (A set is a collection of things and there's no reason those things can't be ... other sets.) And if you have a set that has the emptyset as an element. That set has one element. The empty set.

And that is what the question on Quora is actually asking about. It is asking about the sets $S = \{\emptyset\}$ which has one element. As opposed to $\emptyset = \{\}$ which doesn't have any.

......

I suppose there is a naive confusion about the difference between being a nested element within a set that is an element of a set, with being an element of a set.

If $Beatles = \{John,Paul,George,Ringo\}$ and $FictionalElephants =\{Babar, Tantor, Hathi, Pinkhonkhonk\}$ and $MyFavoriteSets = \{Beatles, FictionalElephants\}$

then how many elements does $MyFavoriteSets$ have?

• It has two: $Beatles$ and $FictionalElephants$.

Is $George$ an element of $MyFavoriteSets$.

• No. $MyFavoriteSets$ has two elements and neither of them are $George$.

But $Beatles$ is an element of $MyFavoriteSets$ and $George$ is an element of $Beatles$ so isn't...?

• No! $George$ is an element of $Beatles$ but $George$ isn't $Beatles$. And it is the sets $Beatles$ and $FictionalElephants$ that are the element of $MyFavoritesSets$; not the elements within those sets.

I suppose another abstraction barrier is that if "" is nothing and $\{\}$ has nothing in it. Then somehow $\emptyset = \{\}$ must also be ... nothing. So $\{\emptyset\}$ must have.... nothing in it because $\emptyset$ is nothing.

This is wrong.

Nothing is ... absence of anything. Making a set and not putting anything in it is making the set. The set is something. Sets are not the things that are in them. They are the sets that collect them. So a set that collects nothing is still a set and it is something. So "" is nothing. $\emptyset = \{\}$ is something although it is something that has nothing inside it. And $\{\emptyset\} = \{\{\}\}$ is something that has something inside it.

Answer 2

The null set, or empty set, contains no elements...

Answer 3

I don't know what the 'null set' is, but if you mean an 'empty set', then the answer is:

the empty set $\emptyset$ contains zero elements: $$|\emptyset| = 0.$$