# well-ordered set

**well-ordered set** A set *S* on which the relation < is defined, satisfying the following properties:

(a) given *x,y,z* in *S*, if *x *< *y* and *y *< *z*, then *x *< *z*

(b) given *x,y* in *S*, then exactly one of the following three possibilities is true: *x *< *y*, *x *= *y*, or *y *< *x*

(c) if *T* is any nonempty subset of *S*, then there exists an element *x* in *T* such that *x* = *y* or *x* < *y*, i.e. *x* ← *y* for all *y* in *T*

This relation < is said to be a *well ordering* of the set *S*.

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