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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.

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    $\begingroup$ Yes they are different. $\{ \emptyset \}$ is a singleton set and $\emptyset$ is null set. $\endgroup$ – Error 404 Mar 12 '17 at 13:28
  • $\begingroup$ Set $\{\varnothing\}$ has an element: $\varnothing$. Set $\varnothing$ has no elements (another notation for this set is $\{\}$). So the sets are different. $\endgroup$ – drhab Mar 12 '17 at 13:29
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    $\begingroup$ 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? $\endgroup$ – user251257 Mar 12 '17 at 13:30
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    $\begingroup$ One of infinitely many: math.stackexchange.com/questions/1951267/…; another would be math.stackexchange.com/questions/1845389/… $\endgroup$ – Asaf Karagila Mar 12 '17 at 13:38
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    $\begingroup$ Yes, the set $A_n$ has $n$ elements; it's the standard representative of the ordinal number $n$ $\endgroup$ – Henno Brandsma Mar 12 '17 at 14:13
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$\emptyset \subset \emptyset$ but $\emptyset \in \{\emptyset\}$ (and also $\emptyset \subset \{\emptyset \}$ since $\emptyset $ is included in every set.)

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

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  • $\begingroup$ That last example is amusing. A set is an element as well as a subset of another set. +1 $\endgroup$ – Paramanand Singh May 21 '17 at 17:19
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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, $|A_i| = i$ $\forall i \in \mathbb{Z}$

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