# Set Theory/Zorn's Lemma and the Axiom of Choice/Well-founded

A binary relation R is well-founded iff for every set A

$A\subseteq R[A]\Rightarrow A=\emptyset$ Theorem: A binary relation R is well-founded iff for every binary relation S

$S\circ R\subseteq R\circ S\Rightarrow R\cap S^{-1}=\emptyset$ Proof: Let R be a well founded relation and let S be a relation such that

$S\circ R\subseteq R\circ S$ Let

$X=field(R)$ and let

$A=dom(R\cap S^{-1})$ Then

$A=dom(R\cap S^{-1})=dom((S\circ R)\cap I_{X})\subseteq dom((R\circ S)\cap I_{X})=dom(S\cap R^{-1})=ran(R\cap S^{-1})\subseteq R[A]$ It follows that A is empty, and therefore $R\cap S^{-1}=\emptyset$ Conversely, suppose that for every relation S we have

$S\circ R\subseteq R\circ S\Rightarrow R\cap S^{-1}=\emptyset$ Let A be a set such that

$A\subseteq R[A]$ Let $B=field(R)$ and let $S=BxA$ . Then

$S\circ R=R^{-1}[B]\times A\subseteq B\times R[A]=R\circ S$ It follows that

$R\circ I_{A}=R\cap (A\times B)=R\cap S^{-1}=\emptyset$ and so

$R[A]=\emptyset$ and consequently $A=\emptyset$ 