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 Í W; A ÃŽ C or A = C^c with C ÃŽ C}

= {A Í W; A Î C or A^c Î C} = C È {C^c; C Î C}.

So that F₁ is closed under complementation.

Next, define the class F₂ as follows:

F₂ = {all finite intersections of members of F₁}

= {A Í W; A = A₁ Ç €¦. Ç 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 Í W; A = A_i with A_i ÃŽ F₂, i = 1,€¦, n, n ‰¥ 1}

= {A Í W; A = A_i with A¹_i Ç €¦ Ç 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).

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i=1