Let G be any countable family of sets, where ϕ ∈ G .Let G 1...

Grimanijd

Grimanijd

Answered

2022-07-06

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.

Answer & Explanation

Charlee Gentry

Charlee Gentry

Expert

2022-07-07Added 19 answers

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 .

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