Difference between revisions of "Infimum"
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: A closely related concept is the [[supremum]], which is the smallest upper bound rather than the greatest lower bound. | : A closely related concept is the [[supremum]], which is the smallest upper bound rather than the greatest lower bound. | ||
==Definition== | ==Definition== | ||
Revision as of 21:27, 19 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])
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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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