Supremum

Mathematical terminology

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This entry is made by Compilation and application of scientific encyclopedia entries of "Science Popularization China" to examine.

The supremum is a aggregate The minimum upper bound of.

The supremum is a concept that is relative to the supremum, which refers to the maximum lower bound of a set.

Chinese name: Supremum
Foreign name: Supremum
Overview: The minimum upper bound of a set

Order theory: The concept of duality is Infimum
Supremum theorem: Basic Proposition of Real Number (Continuity) Theory
application area: Mathematical analysis of order theory

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definition

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Order theory

Supremum is Order theory One of the most basic concepts in.

given Poset ( S , ≤)， A yes S A subset of, then A Of Supremum (also called Minimum upper bound ）sup A Elements defined as meeting the following conditions:

Ⅰ.sup A ∈ S

Ⅱ.∀ a ∈ A ⇒ a ≤ sup A

Ⅲ.∀ a ∈ S , if a Satisfy b ∈ A ⇒ b ≤ a , sup A ≤ a 。

That is: sup A yes A All of upper bound Composed of sets Minimal element (if any).

A The supremum of is also recorded as sup（ A )，lub A ，Lub A Or ∨ A 。

The dual concept of supremum in order theory is Infimum 。

Not all A Can find the supremum.

mathematical analysis

Specific to mathematical analysis Medium. One real number aggregate A , if there is a real number M , making A No more than M , then call it M yes A One of upper bound 。 If there is a minimum upper bound among all those upper bounds, it is called A The supremum of.^[1] That is, there is a real number set A ⊂ R, and the supremum supA of the real number set A is defined as the following number:

（1）

(That is, supA is the upper bound of A)

（2）

(That is, smaller is not the upper bound)

If a number set has an upper bound, it has countless upper bounds; But there is only one supremum, which can be seen intuitively from the meaning of supremum (minimum upper bound). And if a number set has an upper bound, it must have a supremum.^[2]

Common conclusions

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Supremum theorem

In general mathematical analysis textbooks, there are a series of theorems in the chapter of real number theory in order to explain the compactness of real numbers. The former Soviet textbooks with more rigorous theories are generally based on Dedekin's partition theorem Prove other equivalent theorems for the starting point. In order to simplify the teaching materials in China, most of them are proofs based on the supremum theorem. Other theorems that illustrate the continuity of real numbers include Interval nest theorem ， Finite covering theorem wait.^[3]

Supremum theorem It is one of the most basic conclusions in the real number theory and the embodiment of the compactness of real number sets.

Theorem: Any set of non empty real numbers with upper bound (lower bound) must have a supremum (lower bound).