Exercise · ZW-INTERSECTION-OF-FAMILIES-006 · ★★☆ Intermediate
What distinguishes \(x \in \bigcap \mathcal{F}\) from \(x \in \bigcup \mathcal{F}\)?
Choose one option.
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- D. \(\bigcap\) requires belonging to every member, and \(\bigcup\) to some member
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Explanation
The intersection requires membership in every member; the union, in at least one. Confusing them is the most common error with 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
- ZW-INTERSECTION-OF-FAMILIES-001 What does x ∈ bigcap F mean?
- ZW-INTERSECTION-OF-FAMILIES-002 If F = \A, B\, what is bigcap F?
- ZW-INTERSECTION-OF-FAMILIES-003 Which statement is true for every nonempty family F?
- ZW-INTERSECTION-OF-FAMILIES-004 Given F = \\a,b\,\b,c\\, what is bigcap F?
- ZW-INTERSECTION-OF-FAMILIES-005 Let F = \\1,2,3,4\, \2,4,6\, \4,5,6\\. What is bigcap F?
- ZW-INTERSECTION-OF-FAMILIES-007 Let F = \\1,…,k\ : k ∈ \3,5,7\\. What is bigcap F?
- ZW-INTERSECTION-OF-FAMILIES-008 Let F be a nonempty family and G ⊆ F a nonempty subfamily. What relation holds between bigcap F and…
- ZW-INTERSECTION-OF-FAMILIES-009 Let A_n = \x ∈ ℕ : x ≥ n\ for each n ∈ ℕ, with ℕ = \1,2,3,…\, and F = \A_n : n ∈ ℕ\. What is bigcap…