Exercise 1176Logic, sets and proofs › Sets and mapsL1Added 3 Nov 2026

Arbitrary unions and intersections

#arbitrary union#arbitrary intersection#family

Statement

Let $(A_i)_{i\in I}$ be a family of sets. Define $\bigcup_{i\in I}A_i$ and $\bigcap_{i\in I}A_i$ . Compute both for $A_n=[0,1/n]$ with $n\ge1$ .

Hint

$\bigcup$ = elements in at least one $A_i$ ; $\bigcap$ = elements in all $A_i$ .

Show solutionHide solution

Definitions.
- $\bigcup_i A_i=\{x:\exists i\in I,\ x\in A_i\}$ .
- $\bigcap_i A_i=\{x:\forall i\in I,\ x\in A_i\}$ .

Union. $\bigcup_{n\ge1}[0,1/n]=[0,1]$ .

Intersection. $\bigcap_{n\ge1}[0,1/n]=\{0\}$ .

Conclusion. $\bigcup=[0,1]$ , $\bigcap=\{0\}$ .