The meaning of singleton and existence of $\{\varnothing, \{ \varnothing \}\}$

Question

I can't understand difference between $\varnothing$ and $\{\varnothing\}$. What is difference between $x$ and set that has only $x$ as element? I think the concept of set is meaningful when it has more than 1 element. If "$\varnothing$ and $\{\varnothing \}$ is different" is right, then I have another question. When we see the set $\{\varnothing,\{\varnothing\}\}, \varnothing$ and $\{\varnothing\}$ exist simultaneously. It means two empty sets exist. One empty set exists as $\varnothing$ itself and one empty set exists as element of $\{\varnothing\}$. I can't understand this situation. What am i misunderstanding?

Answer 1

Think of sets as bags. $\varnothing$ is an empty bag. $\{\varnothing\}$ is a bag with an empty bag within it. Therefore the outermost bag is not empty: It has a bag inside it.

Also, with this analogy, we have that $\{\varnothing,\{\varnothing\}\}$ a bag is containing an empty bag and a bag with an empty bag within it.

Answer 2

The void set $\varnothing$ has no elements.

The set $\{\varnothing\}$ has one element: the void set $\varnothing$.

The set $\{\varnothing,\{\varnothing\}\}$ has two elements : the void set $\varnothing$ and the set $\{\varnothing\}$.

The key fact is that a set (void or not) is a different thing with respect to his elements and can be element of another set.