## Ordering by inclusion

Consider the sets of natural numbers (partially) ordered by inclusion. Determine the supremum and infimum of the following set: $$M=\{ \{1{,}3,5{,}7,9\}, \{2{,}3,5{,}9\}, \{1{,}5,7{,}9,10\}, \{4{,}5,6{,}9,11\} \}$$.
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 $$M$$.
$$\sup M = \bigcup M = \{1{,}2,…,7{,}9,10{,}11\}$$, $$\inf M = \bigcap M = \{5{,}9\}$$.