Definition:Convex Set (Order Theory)
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This page is about Convex Set in the context of Order Theory. For other uses, see Convex Set.
Definition
Definition 1
A subset $A$ of an ordered set $\struct {S, \preceq}$ is convex (in $S$) if and only if:
- $\forall x, y \in A: \forall z \in S: x \preceq z \preceq y \implies z \in A$
Definition 2
A subset $A$ of an ordered set $\struct {S, \preceq}$ is convex (in $S$) if and only if:
- $\forall x, y \in A: \forall z \in S: x \prec z \prec y \implies z \in A$
Specific Instances
Convex Subset of Natural Numbers
Let $A \subseteq \N$ be a subset of the set of natural numbers.
$A$ is defined as being convex (in $\N$) if and only if:
- $\forall x, y, z \in \N: \paren {x, z \in A \land x \le y \le z} \implies y \in A$
Also see
- Results about convex sets can be found here.