Math 2.0

statement proposition

Uniqueness of the supremum

Statement

Let (P,≤)(P, \le) be a partially ordered set and let SS be a subset of PP. If bb and b′b' are both suprema of SS, then b=b′b = b'.

Remarks

Proof: bb is an upper bound and b′b' is a least upper bound, thus b′≤bb' \le b. In the same way b≤b′b \le b', and antisymmetry gives b=b′b = b'. The same argument shows that a set has at most one infimum.

Explanations

L1 human

A set cannot have two different least upper bounds. If it had two, each one would be at most the other one, so they would be equal.

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Set bounded above, definition, no evidence levelSet bounded abovedefinitionSupremum, definition, no evidence levelSupremumdefinitionUniqueness of the supremum, statement, no evidence levelUniqueness of the sup…statement
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Sources

  1. Web https://en.wikipedia.org/wiki/Infimum_and_supremum, Existence and uniqueness

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