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Least upper bound

Revision as of 07:06, 4 November 2006 by JBL (talk | contribs) (supremum redirects here, and the two terms are equivalent, so no link out)

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Given a subset $S$ in some larger ordered set $R$, a least upper bound or supremum, for $S$ is an element $\displaystyle M \in R$ such that $s \leq M$ for every $s \in S$ and there is no $m < M$ with this same property.

If the least upper bound $M$ of $S$ is an element of $S$, it is also the maximum of $S$. If $M \not\in S$, then $S$ has no maximum.

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