Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Binary endorelation
Preordering relation
Partial ordering relation
Partially ordered set
Minimal element
Minimum element
Set lower bound
Formulation 0
Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that
(i) \begin{equation} X \neq \emptyset \end{equation}
(ii) $E \subseteq X$ is a D78: Subset of $X$
A D2218: Set element $a \in X$ is a lower bound of $E$ in $P$ if and only if \begin{equation} \forall \, x \in E : (a, x) \in {\preceq} \end{equation}
Formulation 1
Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that
(i) \begin{equation} X \neq \emptyset \end{equation}
(ii) $E \subseteq X$ is a D78: Subset of $X$
A D2218: Set element $a \in X$ is a lower bound of $E$ in $P$ if and only if \begin{equation} \forall \, x \in E : a \preceq x \end{equation}
Formulation 2
Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that
(i) \begin{equation} X \neq \emptyset \end{equation}
(ii) $E \subseteq X$ is a D78: Subset of $X$
A D2218: Set element $a \in X$ is a lower bound of $E$ in $P$ if and only if \begin{equation} a \preceq E \end{equation}
Dual definition
» Set upper bound
Child definitions
» D553: Set of lower bounds