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Math Help - Sets

  1. #1
    Super Member Showcase_22's Avatar
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    Sets

    Let A,B,C be sets.

    (i)Which of the following are always true?

    1). A \ ( B \ C) = ( A \ B) U C
    2). A \ (B U C) = (A \ B) \ C
    3). A \ (B <intersection> C)= (A \ B) U (A \ C)
    I have the first one as not always true and the second and third as always true. I just wanted someone to check this because I think i'm doing them wrong.

    Is there a difference between A \ B \ C and (A \ B) \ C?

    Sorry I don't know how to get unions and intersections to work so I used a U for the union and wrote <intersection> for intersections.
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by Showcase_22 View Post
    I have the first one as not always true and the second and third as always true. I just wanted someone to check this because I think i'm doing them wrong.

    Is there a difference between A \ B \ C and (A \ B) \ C?

    Sorry I don't know how to get unions and intersections to work so I used a U for the union and wrote <intersection> for intersections.
    You can rewrite :
    A \backslash B=A \cap B^c

    It's \cup for union and \cap for intersection

    A \ B \ C
    I don't think this is a correct writing, unless you can prove (A\B)\C=A\(B\C)
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  3. #3
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    A\backslash \left( {B\backslash C} \right) = A \cap \left( {B \cap C^c } \right)^c  = A \cap \left( {B^c  \cup C} \right)
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  4. #4
    Super Member Showcase_22's Avatar
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    I don't think this is a correct writing, unless you can prove (A\B)\C=A\(B\C)
    But surely if the set is A not B not C then there would just be A regardless of where you put brackets?

    <br />
A \backslash B=A \cap B^c<br />
    Where did this come from? I'm trying to think of it as a Venn diagram but I can't see what it would look like.

    <br />
A\backslash \left( {B\backslash C} \right) = A \cap \left( {B \cap C^c } \right)^c = A \cap \left( {B^c \cup C} \right)<br />
    ohhh, so A \ B \ C is not the same as (A \ B) \ C. That seems weird! does "\" mean "not"?

    My foundations lecturer introduced "" that apparently also means not. It seems strange that he would introduce two different symbols that apparently mean the same thing! (he also did the same for ^ and  \cap and it's counterpart with the union. Is there a difference between all this notation???)
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  5. #5
    Moo
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    Quote Originally Posted by Showcase_22 View Post
    But surely if the set is A not B not C then there would just be A regardless of where you put brackets?
    Why ?

    Where did this come from? I'm trying to think of it as a Venn diagram but I can't see what it would look like.
    Hmm well, it's the definition oO
    How do you define A \backslash B ?
    It's the set of elements that are in A but not in B, that is to say they are in A and in B^c
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  6. #6
    Super Member Showcase_22's Avatar
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    Hang on, I think i'm missing something.

    "\" means "not" doesn't it?

    For example, "A \ B" means "A not B".

    I'm pretty sure that's what our lecturer told us. =S
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  7. #7
    Moo
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    Quote Originally Posted by Showcase_22 View Post
    Hang on, I think i'm missing something.

    "\" means "not" doesn't it?

    For example, "A \ B" means "A not B".

    I'm pretty sure that's what our lecturer told us. =S
    Hmmm "not A" usually stands for A^c, it affects only one set.
    Here, \ is a binary operator, that is to say it deals with two sets.

    If you prefer, it's "A and not B" and we can refer to
    elements that are in A but not in B
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  8. #8
    Super Member Showcase_22's Avatar
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    Okay then. Going back to the first question:

    A \ (B \ C)=A \ (B \cap C^d) \neq (A \ B) \cup C

    (I used d instead of c since C was already a set).

    The second question would become:

    Need to show: A \ (B \cup C)=(A \ B) \ C
    (A \ B) \ C= A \cap B^d \ C

    Would I need another identity to show that it's true? (if it's true, I think it is). Truthfully, I would much rather remember identities and try to get one side to equal the other side. Venn diagrams are rather limited.
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  9. #9
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    #3 is correct.
    \begin{array}{rcl}<br />
   {A\backslash \left( {B \cap C} \right)} &  =  & {A \cap \left( {B \cap C} \right)^c }  \\<br />
   {} &  =  & {A \cap \left( {B^c  \cup C^c } \right)}  \\<br />
   {} &  =  & {\left( {A \cap B^c } \right) \cup \left( {A \cap C^c } \right)}  \\<br />
   {} &  =  & {\left( {A\backslash B} \right) \cup \left( {A\backslash C} \right)}  \\<br />
   {} & {} & {}  \\<br /> <br />
 \end{array}

    This set operation is known as setminus.
    The are two notations in use: \left( {A\backslash C} \right) = A-C
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