Definition:Lower Bound of Set
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This page is about lower bound in the context of ordered sets. For other uses, see Lower Bound.
Definition
Let $\struct {S, \preceq}$ be an ordered set.
Let $T$ be a subset of $S$.
A lower bound for $T$ (in $S$) is an element $m \in S$ such that:
- $\forall t \in T: m \preceq t$
That is, $m$ precedes every element of $T$.
Subset of Real Numbers
The concept is usually encountered where $\struct {S, \preceq}$ is the set of real numbers under the usual ordering $\struct {\R, \le}$:
Let $\R$ be the set of real numbers.
Let $T$ be a subset of $S$.
A lower bound for $T$ (in $\R$) is an element $m \in \R$ such that:
- $\forall t \in T: m \le t$
Also defined as
Some sources use the terminology the lower bound for the notion of infimum.
Also see
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Orderings
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): partial order
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): partial order