3
$\begingroup$

Suppose that $S$ is a well-ordered set; how can we prove that the following is a total order on the power set of $S$? $$A\prec B\Longleftrightarrow \min(A\triangle B)\in A.$$

1 Answers 1

3

To prove that a relation $<$ is a total order you need to show:

  1. Given $A,B$ exactly one of the $A
  2. $A

The first one: Take two sets $A,B\in P(S)$. Then if $A\neq B$ their symmetric difference $A\triangle B\neq\varnothing$. Thus it has a least element (since $S$ is well ordered). If this element is a member of $A$ then $A

The second one: Let $A