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I am having trouble understanding the difference between these sets.

$\emptyset,\{\emptyset\},\{\{\emptyset\}\},...$

I think they're supposed to be different to eachother, but I read that the empty set is a subset of every set including itself.

So every set in $\emptyset$ is in $\{\emptyset\}$ and every set in $\{\emptyset\}$ is in $\emptyset$. So they're subsets of each other and therefore the same set.

So $\emptyset,\{\emptyset\},\{\{\emptyset\}\},...$ are all the same. Is this right?

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    $\begingroup$ No. $\emptyset$ has no element, while $\{\emptyset\}$ has one element. $\endgroup$ – Berci Mar 16 at 15:47
  • $\begingroup$ Sometimes it helps to think of the empty set as $\{\}$, so $\{\} \in \{\{\}\}$. $\endgroup$ – Hyperion Mar 16 at 15:54
  • $\begingroup$ "So every set in ∅ is in {∅} and every set in {∅} is in ∅." NO; there are no sets in $\emptyset$ while there is exactly one set in $\{ \emptyset \}$. Thus, the two are not the same. $\endgroup$ – Mauro ALLEGRANZA Mar 16 at 16:05
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It is true that every element of the empty set $\varnothing$ is an element of $\{\varnothing\}$ (since the empty set $\varnothing $ has no elements) but it is not true that every element of $\{\varnothing\}$ is an element of $\varnothing$. Indeed, $\varnothing\in \{\varnothing\}$ but $\varnothing\notin \varnothing$. So $\varnothing \subset \{\varnothing\}$ but $\{\varnothing\}$ is not a subset of $\varnothing$.

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$\phi$ simply means the set is empty, i.e has no elements. Eg. A= $\phi$ implies A is an empty set.

On the other hand, {$\phi$} means a set S= {$\phi$} has the element $\phi$ $\Rightarrow$ $\phi \in$ {$\phi$}

Similarly, $\phi \in$ {$\phi$} $\in$ {{$\phi$}}

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It might also help to think of the sets as being boxes, which might contain any number of objects (including infinite). For instance, the set $S = \{\text{apple}, \text{ball}, \text{corkscrew}\}$ might be represented by an actual box containing an apple, a ball, and a corkscrew.

In this analogy, the empty set $A = \varnothing$ is just an empty box. Being the empty set, it is a subset of any other set. That would mean that every item in the empty box is also in every other box. That is manifestly true: There are no items in the empty box that aren't also in every other box.

Note that the empty set is not an element of every other set. For example, the box representing the set $S$ contains three ordinary objects, but does not contain an empty box.

However, something that would contain an empty box would be $B = \{\varnothing\}$, which would be represented by a box that contains only an empty box (which would represent $A$). And $C = \{\{\varnothing\}\}$ would be—you guessed it—a box, which contains only a box, which in turn contains only an empty box.

Observe that these various boxes are evidently not the same: Something that contains only an empty box is not the same thing as an empty box.

(Please don't ask me about $\{\text{box}\}$.)

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