Discussion
Which of the following is the smallest field containing a non-empty subset A of \(\Omega\) ?
Answer: B
The smallest field containing A is {A, AC, φ, Ω}.
A field (algebra) of sets must contain \(\Omega\) and \(\phi\) and be closed under complements and finite unions. Any field containing A therefore contains \(A^{C}\), \(\phi\) and \(\Omega\). These four sets are already closed under complement and union (for instance \(A \cup A^{C} = \Omega\)), so they form a field, and it is the smallest one.
The trivial field \(\{\phi, \Omega\}\) does not contain A unless A is \(\Omega\). The power set is a field containing A, but generally a much larger one, so it is not the smallest.
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