The upper bound is a special element related to a partially ordered set, which refers to the elements in the partially ordered set that are greater than or equal to all elements in its subset.If the number set S is a subset of the real number set R, it obviously has infinite upper bounds, and the smallest upper bound often plays an important role, which is called the supremum of the number set S.[1]
Chinese name
upper bound
Foreign name
upper bound
Nature 1
The upper bound of subsets of partially ordered sets does not necessarily exist
Definition
Elements greater than or equal to all elements in the subset
The upper bound is the same asPosetA special element related to.Refers to an element in a partially ordered set that is greater than or equal to all elements in its subset.Let<A, R>be a partially ordered set,
, if all
If there is xRa, then a is called the upper bound of B in the partially ordered set<A, R>, referred to as the upper bound of B
。If a is the upper bound of B, and there is aRc for any upper bound c of B, then a is called the supremum (or the minimum upper bound) of B, and recorded as
。[2]
Definition on real number set R
Announce
edit
Consider oneSet of real numbersClose M.If there is a real number s, so that no number in M exceeds s, then s is said to be an upper bound of M.
It is expressed by mathematical symbols as follows: for ∀ x ∈ M, there is x ≤ s, then s is called the upper bound of M.
Supremum principle: if the subset M of R has an upper bound, then there must beSupremum；If set M has a lower bound, then there must beInfimum。
Supremum definition: Let S be a number set in R, if the number η∈ R satisfies
(i) For ∀ x ∈ S, η ≥ x, that is, η is the upper bound of S;
(ii) For ∀ a<η, there is x0 ∈ S, so that x0>a, that is, η is the least upper bound of S, then η is called the supremum of number set S;
Definition of supremum: Let S be a number set of R, if the number ξ∈ R satisfies:
(i) For ∀ x ∈ S, ξ ≤ x, that is, ξ is SLower bound；
(ii) For ∀ β>ξ, ∃ x0 ∈ S, so that x0<β, that is, ξ is the greatest lower bound of S, then ξ is called the infimum of S of the number set;[3]
fromDedekin theoremIt is proved that the nonempty number set with upper bound must have a supremum, and the nonempty number set with lower bound must have a supremum.
Let S be a set of non empty upper bounded numbers, that is
Establishment.Access set B is the set of all upper bounds of S, A=R/B. Then:
① It can be known from the selection method
, so
。
, so
, so
。
②
。
③ None of the elements in A is the upper bound of S, ∨
。
Also ∨ Any element in B is the upper bound of S, ∨
。
So there must be
。
According to the Dudekin theorem, either A has a maximum value or B has a minimum value.Let this value be η, and
，
Constant establishment.
Assume that η is the maximum value in A, that is
, so,
。
Also:
，∴
。
But,
, and any element in B is the upper bound of S.
η is the minimum value in B, that is, S has a minimum upper bound (supremum).[1]
give an example
Announce
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To one
, its upper bound may not exist, or there may be more than one.For example, let A={1,2,3}, R={<a, b>| a divide by b}.When Bone=When {2,3}, BoneThere is no upper bound when Btwo={1} When, there is upper bound 1,2,3, and 1 is BtwoThe supremum of.
yes
If the supremum exists, it is unique.When a subset B has an upper bound, it may not have a supremum, and if it has a supremum, it may not be in subset B. For example, for example, a partial ordered set represented by a Hasse graph in an overview graph, with A={a, b, c, d, e} as the basic set, subset B={b, c, d}, with a as the upper bound, a
{b, c, d}. the upper bound and supremum of the subset {e, f} are both f.The subset {c, d, e} has no upper bound and no supremum.[2]
Every finite subset of a nonempty completely ordered set has an upper bound and a lower bound.
For example, 5 is the lower bound of the set {5, 8, 42, 3413934}.
Another example is that for the set {42}, the number 42 is both the upper bound and the lower bound, and all other real numbers are not the upper or lower bound of the set.
Each subset of all natural numbers has a lower bound, because natural numbers have the smallest element (0 or 1, depending on the exact definition of natural numbers).An infinite subset of natural numbers cannot be defined from above.An infinite subset of integers can be defined from below or from above.The infinite subset of rational numbers may or may not be defined from below, or may not be limited to the above.[4]
