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

    \(\sup M = \bigcup M = \{1{,}2,…,7{,}9,10{,}11\}\), \(\inf M = \bigcap M = \{5{,}9\}\).

Difficulty level: Easy task (using definitions and simple reasoning)
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