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



RESET




Word of the Day

29 April 2024

SUBDUCTION

(noun) a geological process in which one edge of a crustal plate is forced sideways and downward into the mantle below another plate


coffee icon

Coffee Trivia

In 1511, leaders in Mecca believed coffee stimulated radical thinking and outlawed the drink. In 1524, the leaders overturned that order, and people could drink coffee again.

coffee icon