Math 2.0

definition

Supremum

Definition

Let (P,≤)(P, \le) be a partially ordered set and let SS be a subset of PP. An element bb of PP is a supremum, or least upper bound, of SS when these two conditions hold:

  1. bb is an upper bound of SS;
  2. if MM is an upper bound of SS, then b≤Mb \le M.

The notation is b=sup⁡Sb = \sup S.

Remarks

A set has at most one supremum, see uniqueness of the supremum. In R\mathbb{R}, sup⁡[0,1)=1\sup [0, 1) = 1 and sup⁡[0,1]=1\sup [0, 1] = 1. The first set does not contain its supremum. In Q\mathbb{Q}, the set {q∈Q:q2<2}\{q \in \mathbb{Q} : q^2 < 2\} is bounded above but has no supremum.

Explanations

L1 human

The supremum is the lowest possible ceiling for a set. For the set {0.9,0.99,0.999,… }\{0.9, 0.99, 0.999, \dots\}, every number at least 11 is a ceiling, and 11 is the lowest of these, so the supremum is 11.

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Sources

  1. Web https://en.wikipedia.org/wiki/Infimum_and_supremum, Formal definition

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