Let W be an abstract set, and let C be an arbitrary nonempty class of subsets of W Define the class F 1 to consists of all members of C as well as all of their complements i e , F 1 A W A C or A C c with C C A W A C or A c C C C c C C So that F 1 is closed under complementation Next, define the class F 2 as follows F 2 all finite intersections of members of F 1 A W A A 1 A m , A i F 1 , i 1,, m, m 1 Also, define the class F 3 by F 3 all finite unions of members of F 2 A W A A i with A i F 2 , i 1,, n , n 1 A W A A i with A 1 i A i mi , A 1 i , , A i mi F1, , m i 1 integers, i 1, , n, n 1 Set F 3 F and show that (i) F is a field (ii) F is the field generated by C i e , F F ( C ) i 1

Question: Let W be an abstract set, and let C be an arbitrary nonempty class of subsets of W. Define the class F 1 to consists

Let W be an abstract set, and let C be an arbitrary nonempty class of subsets of W. Define the classF1 to consists of all members of C as well as all of their complements; i.e.,

F₁ = {A Ã W; A ÃŽ C or A = C^c with C ÃŽ C}

= {A Ã W; A ÃŽ C or A^c ÃŽ C} = C Ãˆ {C^c; C ÃŽ C}.

So that F₁ is closed under complementation.

Next, define the class F₂ as follows:

F₂ = {all finite intersections of members of F₁}

= {A Ã W; A = A₁ Ã‡ €¦. Ã‡ A_m, Ai ÃŽ F₁, i = 1,€¦, m, m ‰¥ 1}.

Also, define the class F₃ by

F₃ = {all finite unions of members of F₂}

= {A Ã W; A = A_i with A_i ÃŽ F₂, i = 1,€¦, n, n ‰¥ 1}

= {A Ã W; A = A_i with A¹_i Ã‡ €¦ Ã‡ A_i^mi,

A¹_i, €¦, A_i^mi ÃŽ F1, , m_i ‰¥ 1 integers,

i = 1, €¦, n, n ‰¥ 1}.

Set F₃ = F and show that

(i) F is a field.

(ii) F is the field generated by C; i.e., F = F(C).

i=1

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