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Thread: Very simple question

  1. #1
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    Very simple question

    For the notation $\displaystyle x \in A$,

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

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

    Take a different example:

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

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

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

    However, the statement $\displaystyle \forall x, x \in A \Rightarrow x \in A \cup B$ 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 $\displaystyle x \in A$,
    Say $\displaystyle x \in A \Rightarrow x \in A \cup B$
    Does $\displaystyle x \in A$ denote all element $\displaystyle x$ of $\displaystyle A$?
    Take a different example:
    Say $\displaystyle x \in A \Rightarrow x \in A - B$
    Does $\displaystyle x \in A$ denote only some element $\displaystyle x$ of $\displaystyle A$?
    How do you decide whether it's some $\displaystyle x $ or all $\displaystyle x$?
    The set of all elements in $\displaystyle A$ is $\displaystyle \{x:x\in A\}$

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

    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 $\displaystyle x \in A \Rightarrow x \in A \cup B$ 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; Feb 18th 2010 at 02:26 PM.
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  5. #5
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    Quote Originally Posted by Plato View Post
    The set of all elements in $\displaystyle A$ is $\displaystyle \{x:x\in A\}$

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

    Now what is the question?
    Plato,

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

    Thanks.
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