SUPREMUM

Etymology

Noun

supremum (plural suprema)

(set theory) (real analysis): Given a subset X of R, the smallest real number that is ≥ every element of X; (order theory): given a subset X of a partially ordered set P (with partial order ≤), the least element y of P such that every element of X is ≤ y.

Usage notes

• Commonly denoted sup(X).

• The supremum of X may not exist, and, if it does, may not be an element of X.

• (order theory)

Formally: Let \(S =\{ t : t \in P : \forall x\in X, x \le t \}\) be the set of upper bounds of X. Then sup(X), if it exists, is the element \(s\in S: \forall y\in S, s \le y\).

The concept of supremum is closely related to the function ∨ (called join). The supremum of two elements, denoted \(\sup\{x,y\}\) can also be written \(x\lor y\). The supremum of a set may be denoted \(\sup(X)\) or \(\bigvee X\).

Synonyms

• (element of a set greater than or equal to all members of a given subset): least upper bound, LUB, sup

Coordinate terms

• infimum

Anagrams

• supermum

Source: Wiktionary



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