# least upper bound

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## least upper bound

[¦lēst ¦əp·ər ′bau̇nd] (mathematics)

The least upper bound of a subset

*A*of a set*S*with ordering < is="" the="" smallest="" element="" of="">*S*which is greater than or equal to every element of*A*. Abbreviated lub. Also known as supremum (sup).## least upper bound

(theory)(lub or "join", "supremum") The least upper bound of
two elements a and b is an upper bound c such that a <= c and
b <= c and if there is any other upper bound c' then c <= c'.
The least upper bound of a set S is the smallest b such that
for all s in S, s <= b. The lub of mutually comparable
elements is their maximum but in the presence of incomparable
elements, if the lub exists, it will be some other element
greater than all of them.

Lub is the dual to greatest lower bound.

(In LaTeX, "<=" is written as \sqsubseteq, the lub of two elements a and b is written a \sqcup b, and the lub of set S is written as \bigsqcup S).

Lub is the dual to greatest lower bound.

(In LaTeX, "<=" is written as \sqsubseteq, the lub of two elements a and b is written a \sqcup b, and the lub of set S is written as \bigsqcup S).

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