First, the difference between the upper and lower limits:
In mathematics, especially in order theory, in some subsets S of posets (k, ≤), the elements of each element whose k is greater than or equal to S are called upper bounds. And the lower bound is defined as each element of k that is less than or equal to S.
1, upper bound: a special element related to a poset, which refers to an element in a poset that is greater than or equal to all elements in its subset.
2. Lower bound: There is a real number A and a real number set B, which pair? X∈B, all have x≥a, then a is called the lower bound of B.
Second, the difference between the upper and lower bounds:
1, the supremum is the minimum upper bound of a set.
If the number set S is a subset of the real number set R and has an upper bound, obviously it has infinite upper bounds, and the minimum upper bound often plays an important role, which is the so-called supremum of the number set S.
2. The lower supremum is a concept that is relatively homogeneous with the upper supremum, which refers to the maximum lower bound of a set.
Third, the difference between upper bound and definite bound:
Neither the upper bound nor the supremum necessarily exists. If both exist, the upper bound is not necessarily unique, but the supremum must be unique.
Fourth, the difference between lower bound and supremum:
Neither the lower bound nor the lower supremum necessarily exist. If both exist, the lower bound is not necessarily unique, but the lower supremum must be unique.
Extended data:
Definition of supremum and supremum
Definition of supremum: let s be a set of numbers in r, if the number η∈R is satisfied.
1 right? X∈S, where η≥x, that is, η is the upper bound of S;
2. Is that right? A < η, x0∈S exists, so that x0 >: A, that is, η is the minimum upper bound of s, then η is called the supremum of several sets s;
Definition of supremum: Let s be a set of numbers of r, if the number ξ∈R satisfies:
1 right? X∈S has ξ≤x, that is, ξ is the lower bound of S;
2. Is that right? β& gt； ξ,? X0∈S, so X0
It is proved by Dai Dejin's theorem that a nonempty set with an upper bound must have an upper bound and a nonempty set with a lower bound must have a lower bound.
References:
Baidu Encyclopedia-Shangquejie
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Baidu encyclopedia-knight errant
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Baidu encyclopedia-lower bound
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Baidu encyclopedia-upper limit