Difference between revisions of "Infimum"

From Maths
Jump to: navigation, search
(Created page with "{{Stub page|Needs fleshing out, INCOMPLETE PAGE}} : A closely related concept is the supremum, which is the smallest upper bound rather than the greatest lower bound. ==De...")
 
m
Line 7: Line 7:
 
# {{M|1=\forall a\in A[\text{inf}(A)\le a]}} (that {{M|\text{inf}(A)}} is a [[lower bound]])
 
# {{M|1=\forall a\in A[\text{inf}(A)\le a]}} (that {{M|\text{inf}(A)}} is a [[lower bound]])
 
# {{M|1=\forall x\in\underbrace{\{y\in X\ \vert\ \forall a\in A[y\le a]\} }_{\text{The set of all lower bounds} }\ \ [\text{inf}(A)\ge x]}} (that {{M|\text{inf}(A)}} is an [[upper bound]] of all lower bounds of {{M|A}})
 
# {{M|1=\forall x\in\underbrace{\{y\in X\ \vert\ \forall a\in A[y\le a]\} }_{\text{The set of all lower bounds} }\ \ [\text{inf}(A)\ge x]}} (that {{M|\text{inf}(A)}} is an [[upper bound]] of all lower bounds of {{M|A}})
===For subsets of the real numbers===
+
#* '''Claim 1: ''', this is the same as {{M|1=\forall x\in X\exists a\in A[x>\text{inf}(A)\implies a<x]}}<ref group="Note">This would require {{M|A\ne\emptyset}}</ref><ref group="Note">Let some {{M|x\in X}} be given, if {{M|x\le\text{inf}(A)}} we can choose any {{M|a\in A}} as for [[implies]] if the LHS of the {{M|\implies}} isn't true, it matters not if we have the RHS or not.</ref>
 +
==Proof of claims==
 +
{{Requires proof|Make a subpage and put the proof here}}
 +
==See also==
 +
* [[Passing to the infimum]]
 +
* [[Supremum]]
 +
** [[Passing to the supremum]]
 +
==Notes==
 +
<references group="Note"/>
 
==References==
 
==References==
 
<references/>
 
<references/>
 
{{Order theory navbox|plain}}
 
{{Order theory navbox|plain}}
 
{{Definition|Order Theory|Real Analysis|Set Theory}}
 
{{Definition|Order Theory|Real Analysis|Set Theory}}

Revision as of 01:12, 14 April 2016

(Unknown grade)
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Needs fleshing out, INCOMPLETE PAGE
A closely related concept is the supremum, which is the smallest upper bound rather than the greatest lower bound.

Definition

An infimum or greatest lower bound (AKA: g.l.b) of a subset [ilmath]A\subseteq X[/ilmath] of a poset [ilmath](X,\preceq)[/ilmath][1]:

  • [ilmath]\text{inf}(A)[/ilmath]

such that:

  1. [ilmath]\forall a\in A[\text{inf}(A)\le a][/ilmath] (that [ilmath]\text{inf}(A)[/ilmath] is a lower bound)
  2. [ilmath]\forall x\in\underbrace{\{y\in X\ \vert\ \forall a\in A[y\le a]\} }_{\text{The set of all lower bounds} }\ \ [\text{inf}(A)\ge x][/ilmath] (that [ilmath]\text{inf}(A)[/ilmath] is an upper bound of all lower bounds of [ilmath]A[/ilmath])
    • Claim 1: , this is the same as [ilmath]\forall x\in X\exists a\in A[x>\text{inf}(A)\implies a<x][/ilmath][Note 1][Note 2]

Proof of claims

(Unknown grade)
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
Make a subpage and put the proof here

See also

Notes

  1. This would require [ilmath]A\ne\emptyset[/ilmath]
  2. Let some [ilmath]x\in X[/ilmath] be given, if [ilmath]x\le\text{inf}(A)[/ilmath] we can choose any [ilmath]a\in A[/ilmath] as for implies if the LHS of the [ilmath]\implies[/ilmath] isn't true, it matters not if we have the RHS or not.

References

  1. Lattice Theory: Foundation - George Grätzer