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Math Help - Very simple question

  1. #1
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    Very simple question

    For the notation x \in A,

    Say x \in A \Rightarrow x \in A \cup B

    Does x \in A denote all element x of A?

    Take a different example:

    Say x \in A \Rightarrow x \in A - B

    Does x \in A denote only some element x of A?

    How do you decide whether it's some x or all x?
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  2. #2
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    By itself, the statement x \in A \Rightarrow x \in A \cup B is meaningless.

    However, the statement \forall x, x \in A \Rightarrow x \in A \cup B does make sense.

    Implications such as this only make sense with universal quantifiers.
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  3. #3
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    Quote Originally Posted by novice View Post
    For the notation x \in A,
    Say x \in A \Rightarrow x \in A \cup B
    Does x \in A denote all element x of A?
    Take a different example:
    Say x \in A \Rightarrow x \in A - B
    Does x \in A denote only some element x of A?
    How do you decide whether it's some x or all x?
    The set of all elements in A is \{x:x\in A\}

    The set \{x:x\in (A\setminus B)\} is the set of all x belonging to A but not to B.

    Now what is the question?
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  4. #4
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    Quote Originally Posted by icemanfan View Post
    By itself, the statement x \in A \Rightarrow x \in A \cup B is meaningless.
    It is not meaningless. It says "If x is in A then x is in A union B".
    That is perfectly good well formed English sentence. And it is true.
    Last edited by Plato; February 18th 2010 at 03:26 PM.
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  5. #5
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    Quote Originally Posted by Plato View Post
    The set of all elements in A is \{x:x\in A\}

    The set \{x:x\in (A\setminus B)\} is the set of all x belonging to A but not to B.

    Now what is the question?
    Plato,

    You have answered my question pointedly by showing the set builder notation.

    Thanks.
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