Upper bound - Imported Knowlpedia Content

An upper bound of a subset $A$ of a partially ordered set $(P,\le)$ is an element $u\in P$ such that $a\le u$ for all $a\in A$ . The subset $A$ is bounded above if it has at least one upper bound in $P$ .

Remarks

Upper bounds are paired with lower bounds and are used to define least upper bounds (the supremum in real analysis).

Examples

In $(\mathbb{Z},\le)$ , the integer $10$ is an upper bound for the subset $\{1,4,7\}$ .
In the poset $(\mathcal P(X),\subseteq)$ , the set $A\cup B$ is an upper bound for $\{A,B\}$ , where $A\cup B$ is the union of $A$ and $B$ .