Some questions about sets

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Question from a parent:

Hello,

(I) - In set theory can a given set contain both elements and subsets, as "elements", as in :
A = {1,2,{},3,{{3}},8}

If yes :
1 - then, is |A| = 6 ?
2 - if the empty set is a subset of every set,
2. 1. does {} = {{}}, {{{}}} etc? , and if the is true what are the respective cardinals of the latter three? (0,1,1?).
2 . 2. Why isn't {1} equal to {{},1}? and why should these two be equal without having the same cardinality?

Sorry if my questions are not well expressed.

Thank you for the great service you provide.

Thanks for the kind words. Your questions are expressed perfectly.

\[A = \left\{1,2,\{ \; \}, 3, \{\{3\}\},8 \right\}\]

is a legitimate set and you are correct $|A| = 6.$

{} is not equal to {{}} since {} has no elements and {{}} has one element, the set {}. Neither of these is equal to {{{}} since {{{}}} has one element which is {{}}. Hence |{}| = 0. |{{}}| = 1 and|{{{}}}|= 1.

You have answered your question about {1} and {{}, 1} yourself. |{1}| = 1 and |{{}, 1}|=2 so they can't be equal. Another way to see this is that {} is an element of {{}, 1} but it's not an element of {1}.

Write back if we can be of further assistance,
Harley

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