



 
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, 



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