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/.

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