Exercise · ZW-UNION-OF-FAMILIES-005 · ★☆☆ Basic
For any family \(\{A_i\}_{i \in I}\), which statement is true?
Choose one option.
Show answer
- B. \(A_j \subseteq \bigcup_{i \in I} A_i\) for every \(j \in I\)
Explanation
Each \(A_j\) is contained in the union, because all its elements belong to at least one set in the family. The reverse fails in general, as the \(A_1, A_2, A_3\) example shows.
Review the concept: Union of families
The union of a family \(\{A_i\}_{i \in I}\) is the set of objects that belong to at least one of the \(A_i\).
\[\bigcup_{i \in I} A_i = \{x \mid \exists\, i \in I,\ x \in A_i\}\]
More practice problems: union of families
- ZW-UNION-OF-FAMILIES-001 What does x ∈ bigcup_i ∈ I A_i mean?
- ZW-UNION-OF-FAMILIES-002 Let A_1 = \1,2\, A_2 = \2,3\ and A_3 = \3,4\. What is bigcup_i=1^3 A_i?
- ZW-UNION-OF-FAMILIES-003 With the same A_1, A_2, A_3 as before, what is bigcap_i=1^3 A_i?
- ZW-UNION-OF-FAMILIES-004 What is bigcup_i ∈ ∅ A_i for a family indexed by the empty set?
- ZW-UNION-OF-FAMILIES-006 If I = \1, 2\, which set is bigcup_i ∈ I A_i?
- ZW-UNION-OF-FAMILIES-007 Let A_n = \n, n+1\ for n ∈ \1,2,3\. How many elements does bigcup_n A_n have?
- ZW-UNION-OF-FAMILIES-008 Let A_n = \n\ for n ∈ \1, 2, dots, 10\. What is bigcup_n A_n?
- ZW-UNION-OF-FAMILIES-009 Let F = \\1,2\, \2,3\\, a family of two sets. What is bigcup F?