Upper bound

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 $P$.

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 $B$.