Difference between revisions of "Infimum"
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# {{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}}) | ||
| − | === | + | #* '''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
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- 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:
- [ilmath]\forall a\in A[\text{inf}(A)\le a][/ilmath] (that [ilmath]\text{inf}(A)[/ilmath] is a lower bound)
- [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])
Proof of claims
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Make a subpage and put the proof here
See also
Notes
- ↑ This would require [ilmath]A\ne\emptyset[/ilmath]
- ↑ 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
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