Compliments Proof 3

Let X \subset Y \subset Z . Prove that C_{Y}(X) \subset C_{Z}(X) .

In order for C_{Y}(X) \subset C_{Z}(X), \forall_{x \in C_{Y}(X)}(x \in C_{Z}(X)) \Rightarrow \exists_{x \notin X} : x \in Y \cup Z because X \subset Y \subset Z . If we assume that \forall_{x \in X}(x \in Y \cup Z) then C_{Y}(X) and C_{Z}(X) would both be \emptyset and \emptyset \not\subset \emptyset . \therefore \exists_{x \notin X} : x \in Y \cup Z . Also \not\exists_{x \in X}: x \notin Y \cup Z because then X \not\subset Y which creates a contradiction with the axiom above. Since \exists_{x \in Y \cup Z}: x \notin X , \forall_{x \in C_{Y}(X)}(x \in C_{Z}(X)) \Rightarrow C_{Y}(X) \subset C_{Z}(X) .

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