Let G be any countable family of sets, where ϕ ∈ G .Let G 1 be the family of all finite union of difference of sets in G .Let G 2 be the family of all finite union of difference of sets in G 1 .May be I am misunderstanding the meaning of "finite union of difference" but to me here G 1 = G 2 . However my text book says it is not necessary. How so?This is used in the textbook to prove the ring generated by countable generator is countable.

Question

Let       G   be any countable family of sets, where   ϕ  ∈      G  .Let             G        1   be the family of all finite union of difference of sets in       G  .Let             G        2   be the family of all finite union of difference of sets in             G        1  .May be I am misunderstanding the meaning of &quot;finite union of difference&quot; but to me here             G        1    =            G        2  . However my text book says it is not necessary. How so?This is used in the textbook to prove the ring generated by countable generator is countable.

Charlee Gentry · Accepted Answer

Let       G    =  {  ∅  ,  {  1  }  ,  {  2  }  ,  {  1  ,  2  ,  3  ,  4  }  }.Then             G        1    =  {  ∅  ,  {  1  }  ,  {  2  }  ,  {  1  ,  2  }  ,  {  2  ,  3  ,  4  }  ,  {  1  ,  3  ,  4  }  ,  {  1  ,  2  ,  3  ,  4  }  }.Now   {  3  ,  4  }  =  {  2  ,  3  ,  4  }  ∖  {  2  } is an element of             G        2  , but   {  3  ,  4  }  ∉            G        1  .