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I have a set with

{1, {1}, {1,1}}

I was wondering whether the power set of this set would be

{null, 1, {1}, {1, 1}, {1, {1}, {{1}, {1, 1,}}, {1, {1, 1}}, {1, {1}, {1, 1}}} 

or just

{null, 1}

I would think it would be the first because the sets are distinct, but I also recognize that the sets are essentially the same meaning that the second option would also make sense.

I think the main issue that I am having with this is differentiating between cardinality and distintic sets. The question that would naturally arise in my mind would be if sets with the same cardinality and the same elements make that set the same.

Thanks for any help you can offer.

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  • $\begingroup$ A set is determined by its elements. Consequently the sets $\{1\}$ and $\{1,1\}$ are the same. It is the unique set that contains only $1$ as element. $\endgroup$ – drhab Oct 22 '18 at 18:01
  • $\begingroup$ At the same time, note that curly braces demand attention - $\{\{1\}\}$ is very much not the same as $\{1\}$. $\endgroup$ – Noah Schweber Oct 22 '18 at 18:10
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Essential is that sets are determined by their elements. This on base of the axiom of extensionality.

Consequently if $A=\{1,\{1\},\{1,1\}\}$ then also $A=\{1,\{1\}\}$.

Then: $$\wp(A)=\{\varnothing,\{1\},\{\{1\}\},\{1,\{1\}\}\}$$

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