> [!definition] > > $ > \inf S = b \in S \quad \text{where } b \ge x \forall x \in A, > b \le x \forall x \in \{\text{Upper Bounds of A}\} > $ > An element of a [[Partial Order|partially ordered]] [[Set|set]] $b \in S$ is the **supremum** (*least* upper bound) of a subset $A \subseteq S$ if: > - $b$ is an upper bound of $A$. > - $b$ is a lower bound of the set of all upper bounds of $A$. (For any upper bound $x$ for $A$, $b \le x$). > > If the supremum exists, then it is unique. If the supremum is in the set itself, then it is the maximum. > [!definition] For Extended Sets > > For empty sets, $\sup \emptyset = \infty$ and $\inf \emptyset = -\infty$. > > If the set is unbounded from above, then $\sup S = \infty$. If it is unbounded from below, then $\inf S = -\infty$. > [!definition] > > $ > \sup A = b \in S \quad \text{where } b \le x \forall x \in A, > b \ge x \forall x \in \{\text{Lower Bounds of A}\} > $ > An element of $S$ is the **infimum** *greatest* lower bound of $A$ if: > - $b$ is a lower bound of $A$. > - $b$ is an upper bound of the set of all lower bounds of $A$. (For any lower bound $x$ for $A$, $b \ge x$). > > If the infimum exists, then it is unique. If the infimum is in the set itself, then it is the minimum.