Prove that 7 5 3 is a well ordered set Note Obviously this s

Prove that {-7, -5, -3,.....} is a well ordered set.

Note: Obviously this set has least element, namely -7, but need to show all subsets of this set have a least element too in order to be W.O. It makes since that they will all have least element since the original set contains least just not sure how to go about proof.

Solution

Solution :-

Definition of Well Ordered Set :- A well order on a set S is a total order on a set S is a total order on S with the property that every non empty subset of S has least element in this ordering.

The set S together with well-order relation is called as a well ordered set.

Here we have given Set S = {-7,-5,-3,....}

Here -7 is the least element in this ordering.

The set of numbers can be written as S = { -7+2n | 0 n < w }

Therefore the set S is well ordered set.