# Definition:Well-Ordering/Definition 1

## Definition

Let $\struct {S, \preceq}$ be an ordered set.

The ordering $\preceq$ is a well-ordering on $S$ if and only if every non-empty subset of $S$ has a smallest element under $\preceq$:

$\forall T \subseteq S, T \ne \O: \exists a \in T: \forall x \in T: a \preceq x$

## Also see

• Results about well-orderings can be found here.