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Set Theory Question

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  1. Oct 11, 2016 #1
    1. The problem statement, all variables and given/known data
    ##C \subseteq A \cap B \implies A \cap B \cap C = C##

    2. Relevant equations
    How do I get rid of the "belongs to" term on the right hand side? I know I need to prove either the left hand or the right hand side of the "or" term is correct, I'm just not sure how to get there.

    3. The attempt at a solution
    ~##(C \subseteq A \cap B) \cup (A \cap B \cap C = C)##

    right hand side (right of the "or"):
    ##C \subseteq A \cap B \cap C## (Trivial)
    ##A \cap B \cap C \subseteq C## (This is the one we want to prove)

    So all together:

    ~##(C \subseteq A \cap B) \cup (A \cap B \cap C \subseteq C)##
    ##\exists x \in C \therefore x \in A \cap B)##
    ##(\sim a \cup \sim b) \cup (a \cap b \cap c \subseteq C)##
     
  2. jcsd
  3. Oct 11, 2016 #2

    fresh_42

    Staff: Mentor

    I don't really understand your complexity here. Can't you simply use ##(X \subseteq Y) \wedge (Y \subseteq X) \Longrightarrow X = Y ##?
     
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