How to Prove Well-Ordered Set?

A set of real numbers is said to be well-ordered if every nonempty subset in it has a smallest element. A well-ordered set must be nonempty and have a smallest element. Having a smallest element does not guarantee that a set of real numbers is well-ordered.

What makes a set well-ordered?

In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set.

Which of the following set have well-ordering property?

A nonempty subset S of R is well–ordered if every non-empty subset of S has a smallest element. The Well-Ordering Principle: The set N is well-ordered. Example. The following sets are well-ordered: (1) N ∪ {0} (2) N ∪ {−1,0} (3) N ∪ {−3,−2,−1} (4) {n ∈ N : n > 5} • Example.