Ordering by inclusion
Task number: 2826
Consider the sets of natural numbers (partially) ordered by inclusion. Determine the supremum and infimum of the following set: \( M=\{ \{1{,}3,5{,}7,9\}, \{2{,}3,5{,}9\}, \{1{,}5,7{,}9,10\}, \{4{,}5,6{,}9,11\} \} \).
Resolution
An upper bound of the set \(M\) is any set of numbers that contains all of the sets in \(M\) as subsets. The smallest such upper bound is the union of the sets in \(M\).
A lower bound of the set \(M\) is any set of numbers, that is contained in all sets in \(M\) as a subset. The largest such lower bound is the intersection of the sets in \(M\).
Result
\(\sup M = \bigcup M = \{1{,}2,…,7{,}9,10{,}11\}\), \(\inf M = \bigcap M = \{5{,}9\}\).
