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  1. #1
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    set question

    A,B,C are sets with A union B = B intersection C. The question asks what do these operations imply about A and C. I let x be an element satisfying the equation and found that any element in A must be in B and any element in B must be in C. Does this make sense and is there anything else I can say about A and C?
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    Quote Originally Posted by PvtBillPilgrim View Post
    A,B,C are sets with A union B = B intersection C. The question asks what do these operations imply about A and C. I let x be an element satisfying the equation and found that any element in A must be in B and any element in B must be in C. Does this make sense and is there anything else I can say about A and C?
    Did you try drawing a picture?
    I get that $\displaystyle A\subseteq B\subseteq C$.
    But we need to be careful, pictures can decieve us.

    See if you can prove the above statement.
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  3. #3
    Moo
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    Hello,
    Quote Originally Posted by PvtBillPilgrim View Post
    A,B,C are sets with A union B = B intersection C. The question asks what do these operations imply about A and C. I let x be an element satisfying the equation and found that any element in A must be in B and any element in B must be in C. Does this make sense and is there anything else I can say about A and C?
    You have proved that any element of A will be in C. This means that $\displaystyle A \subseteq C$

    And this is all you can say about A and C.
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  4. #4
    MHF Contributor Matt Westwood's Avatar
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    Nice question.

    Let $\displaystyle x \in A$.

    Then $\displaystyle x \in A \cup B$ because $\displaystyle A \subseteq A \cup B$ (not sure whether you'd be expected to prove the latter).

    So as $\displaystyle A \cup B = B \cap C$, we have $\displaystyle x \in B \cap C$.

    But because $\displaystyle B \cap C \subseteq C$ (again something you may need to prove separately), it follows that $\displaystyle x \in C$.

    Thus $\displaystyle x \in A \rightarrow x \in C$ and thus $\displaystyle A \subseteq C$.
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  5. #5
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    Thanks for the replies.

    Can you say A is a subset of B and B is a subset of C (therefore A is a subset of C)? Or can you say nothing about B?
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  6. #6
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    Yes, if set A is a subset of set B and set B is the subset of set C, then set A is indeed the subset of set C.

    Well, to make things more interesting than usual, it is not wrong to say that set A is an element of the set B.
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  7. #7
    MHF Contributor Matt Westwood's Avatar
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    Yes it is.

    $\displaystyle A \subseteq B$ does NOT mean that $\displaystyle A \in B$. Sorry, but that is completely wrong.
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