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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.

Asaf Karagila
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Jaideep Khare
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    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
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    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
  • @user251257 I meant looked Wiki of null set etc. – Jaideep Khare Mar 12 '17 at 13:32
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    One of infinitely many: http://math.stackexchange.com/questions/1951267/why-is-emptyset-not-emptyset/1951388; another would be http://math.stackexchange.com/questions/1845389/emptyset-0-emptyset-0-differences/1845392 – Asaf Karagila Mar 12 '17 at 13:38
  • So, answer should be last option, isn't it? – Jaideep Khare Mar 12 '17 at 13:45
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    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

2 Answers2

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

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

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

Surb
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