Exercise · ZW-INTERSECTION-OF-FAMILIES-009 · ★★★ Advanced
Let \(A_n = \{x \in \mathbb{N} : x \geq n\}\) for each \(n \in \mathbb{N}\), with \(\mathbb{N} = \{1,2,3,\ldots\}\), and \(\mathcal{F} = \{A_n : n \in \mathbb{N}\}\). What is \(\bigcap \mathcal{F}\)?
Choose one option.
A \(\varnothing\) B \(\{2,3,4,\ldots\}\) C \(\mathbb{N}\) D \(\{1\}\) Show answer ← Back to the question
Explanation For each \(x\) there is \(n > x\) with \(x \notin A_n\), so no element lies in every set . The intersection is empty, even though each \(A_n\) is infinite.
The intersection of a nonempty family \(\mathcal{F}\) of sets is the set \(\bigcap \mathcal{F}\) of objects that belong to every member of the family.
\[\bigcap \mathcal{F} = \{x \mid \forall A \in \mathcal{F},\ x \in A\}\]
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