# Are set notations $\emptyset$ and $\{\emptyset\}$ Different?

Question : Let $A_0=\emptyset$ (the empty set).For each $i=1,2,3,...,$ define the set -

$A_i=A_{i-1}$ $\cup$ {$A_{i-1}$}. The set $A_3$ is :

• $\emptyset$

• $\{\emptyset\}$

• $\{\emptyset, \{\emptyset\}\}$

• $\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}$

My Work : I Googled this, but didn't get anything useful.

• Yes they are different. $\{ \emptyset \}$ is a singleton set and $\emptyset$ is null set. – Error 404 Mar 12 '17 at 13:28
• Set $\{\varnothing\}$ has an element: $\varnothing$. Set $\varnothing$ has no elements (another notation for this set is $\{\}$). So the sets are different. – drhab Mar 12 '17 at 13:29
• I am curious, how have you googled that? As for your question, think about sets as boxes. Is an empty box different from a box containing an empty box? – user251257 Mar 12 '17 at 13:30
• One of infinitely many: math.stackexchange.com/questions/1951267/…; another would be math.stackexchange.com/questions/1845389/… – Asaf Karagila Mar 12 '17 at 13:38
• Yes, the set $A_n$ has $n$ elements; it's the standard representative of the ordinal number $n$ – Henno Brandsma Mar 12 '17 at 14:13

$\emptyset \subset \emptyset$ but $\emptyset \in \{\emptyset\}$ (and also $\emptyset \subset \{\emptyset \}$ since $\emptyset$ is included in every set.)

Now $$\{\emptyset \}\in \{\emptyset ,\{\emptyset \}\}\quad \text{and}\quad \{\emptyset \}\subset \{\emptyset ,\{\emptyset \}\}.$$

• That last example is amusing. A set is an element as well as a subset of another set. +1 – Paramanand Singh May 21 '17 at 17:19

Yes, they're different.

$\emptyset$ denotes a set with no elements (null set) whereas $\{\emptyset\}$ denotes a set with 1 element, namely, the empty set being the element of this set.

Therefore, $A_i \neq A_j$ $\forall i \neq j$

Also, observe that, $|A_i| = i$ $\forall i \in \mathbb{Z}$