# Compliments Proof

Let $A \subset S$, $B \subset S$. Prove that $A \subset B$ if and only if $C(B) \subset C(A)$

Let $C(B) \subset C(A)$ and assume that $A \not\subset B$. From this assumption it follows that $\exists_{x\in A} : x\notin B$, which means that $\exists_{x\notin B} : x\in A$, which can also be stated as $\exists_{x\in C(B)} : x \notin C(A) \therefore C(B) \not\subset C(A)$ which contradicts with the above assumption and therefore $A \subset B$. Now if we let $A \subset B$ and assume that $C(A) \not\subset C(B)$ than it follows that $\exists_{x\in C(A)} : x \notin C(B) \Rightarrow \exists _{x\in A} : x \notin B$. This means that $A \notin B$ which contradicts the above statement and therefore $C(A) \subset C(B)$. This came from  http://ashleymills.com/.