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Math Help - set theory different approach

  1. #1
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    set theory different approach

    if we are to prove
    X-(X-A) = A,
    what we normally do is for an arbitrary x suppose ,
    x E X-(X-A)
    --> proof
    --> x E A

    and then
    x E A
    --->proof
    ---> x E X-(X-A)
    this tells us X-(X-A) = A,
    its perfect and theres no wrong with it.

    but my question is what happens if I use biconditional statement as
    x E X-(X-A) <---> x E X AND x ~E X -A
    <---> some proof here
    <---> x E A

    then we dont want to split it into two parts as did earlier.

    but why dont we normally prove like this???
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  2. #2
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    X\backslash \left( {X\backslash A} \right) = X \cap \left( {X \cap A^c } \right)^c  = X \cap \left( {X^c  \cup A} \right) = \emptyset  \cup A = A
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  3. #3
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    yeah.. but i just wanted to know whether my approach is correct or wrong...
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  4. #4
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    Quote Originally Posted by doresa View Post
    yeah.. but i just wanted to know whether my approach is correct or wrong...
    Frankly, I cannot follow what you are even trying to say in the OP.
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  5. #5
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    i have mentioned it in my first post.....
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  6. #6
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    Quote Originally Posted by doresa View Post
    i have mentioned it in my first post.....
    I meant that your first post is impossible to follow.
    I have no idea what you are doing with that.
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  7. #7
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    suppose i have to prove X-(X-A) = A,

    1st step :
    for an arbitrary x suppose ,
    x E X-(X-A)
    --> proof goes here
    --> x E A
    and 2nd step:
    x E A
    --->proof goes here
    ---> x E X-(X-A)
    this is the normal structure of proving this sum.ok?

    but if i do it like this...using the definition of set difference,
    x E X-(X-A) <---> x E X AND x ~E X -A
    <---> some proof here
    <---> x E A
    is it wrong?
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  8. #8
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    O.K. That is what is usually called a 'pick-a-point proof'.
    Some instructors perfer that all proofs be done that way.
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  9. #9
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    x'cuse me
    Some instructors perfer that all proofs be done that way.
    here that way means what way?
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  10. #10
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    and why dont you normally use the latter method?
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