Definition:Greatest
Jump to navigation
Jump to search
Definition
Greatest Element of Ordered Set
Let $\struct {S, \preceq}$ be an ordered set.
An element $x \in S$ is the greatest element (of $S$) if and only if:
- $\forall y \in S: y \preceq x$
That is, every element of $S$ precedes, or is equal to, $x$.
The Greatest Element is Unique, so calling it the greatest element is justified.
Thus for an element $x$ to be the greatest element, all $y \in S$ must be comparable to $x$.
Greatest Set
Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$.
Let $\struct {\TT, \subseteq}$ be the ordered set formed on $\TT$ by the subset relation.
Then $T \in \TT$ is the greatest set of $\TT$ if and only if $T$ is the greatest element of $\struct {\TT, \subseteq}$.
That is:
- $\forall X \in \TT: X \subseteq T$