Exercise 7.76

Exercise 7.76#

  1. Explain why [2, 6] ∈ \(π‘œ_{[0,8]}\)

  2. Explain why [2, 6] βˆ‰ \(π‘œ_{[0,5]} βˆͺ π‘œ_{[4,8]}\)

Alternative answer#

Notice that \(o_{[a,b]}\) is an infinite set of finite closed intervals, not just a finite closed interval. When we form \(o_{[0,5]} βˆͺ o_{[4,8]}\) we’ll only β€œdeduplicate” all the closed intervals in \(o_{[4,5]}\) (roughly speaking).

Author’s solution#

We have \(π‘œ_{[π‘Ž,𝑏]} := \{[𝑑, 𝑒] ∈ 𝕀ℝ \enspace | \enspace π‘Ž < 𝑑 ≀ 𝑒 < 𝑏\}\).

  1. Since \(0 ≀ 2 ≀ 6 ≀ 8\), we have \([2, 6] ∈ π‘œ_{[0,8]}\) by the above formula.

  2. In order to have \([2, 6] ∈^? π‘œ_{[0,5]} βˆͺ π‘œ_{[4,8]}\), we would need to have either \([2, 6] ∈^? π‘œ_{[0,5]}\) or \([2, 6] ∈^? π‘œ_{[4,8]}\). But the formula does not hold in either case.