Exercise · ZW-INTERSECTION-OF-FAMILIES-008 · ★★☆ Intermediate

Let \(\mathcal{F}\) be a nonempty family and \(\mathcal{G} \subseteq \mathcal{F}\) a nonempty subfamily. What relation holds between \(\bigcap \mathcal{F}\) and \(\bigcap \mathcal{G}\)?

Choose one option.

Show answer
  • B. \(\bigcap \mathcal{F} \subseteq \bigcap \mathcal{G}\)

Explanation

An element common to all members of \(\mathcal{F}\) is also common to the members of the subfamily \(\mathcal{G}\). By requiring fewer memberships in \(\mathcal{G}\), its intersection can be larger.

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Review the concept: Intersection of families

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.

\[\bigcap \mathcal{F} = \{x \mid \forall A \in \mathcal{F},\ x \in A\}\]

More practice problems: intersection of families

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