Exercise · ZW-VITALI-SET-002 · ★☆☆ Basic
What equivalence relation is used to construct the Vitali set?
Choose one option.
Show answer
- B. \(x\sim y\) if \(x-y\in\mathbb{Q}\)
Explanation
The relation identifying two reals when their difference is rational produces disjoint classes. The other relations are either not equivalence relations or do not generate the right partition.
Review the concept: Vitali set
The Vitali set is a subset of \([0,1]\) containing exactly one representative of each class of \(\mathbb{R}\) modulo \(\mathbb{Q}\), and it is not Lebesgue measurable.
\[x \sim y \iff x - y \in \mathbb{Q}\]
More practice problems: vitali set
- ZW-VITALI-SET-001 Which axiom is used to choose a representative from each equivalence class in Vitali's construction?
- ZW-VITALI-SET-003 Which property of Lebesgue measure is used in Vitali's contradiction?
- ZW-VITALI-SET-004 What is the Lebesgue measure of the interval [0,1]?
- ZW-VITALI-SET-005 How many rationals in [-1,1] are used to translate the Vitali set?
- ZW-VITALI-SET-006 If V is a Vitali set and q∈ℚ, what is the Lebesgue measure of V+q when m(V)=0?
- ZW-VITALI-SET-007 If V is a Vitali set, what does the union bigcup_n (V+q_n) contain for an enumeration q_n of ℚ∩[-1,…
- ZW-VITALI-SET-008 If m(V)>0 and the translates V+q_n are disjoint, what follows about the measure of their union?
- ZW-VITALI-SET-009 What is the error in claiming the Cantor set is non-measurable because it is uncountable?