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Math Help - algebraic laws of sets

  1. #1
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    Exclamation algebraic laws of sets

    Excuse me,how to do this type of question?

    Using the laws of the algebra of sets show that

    (a-b)∩(a∩b)'=a∩b'
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  2. #2
    Super Member Deadstar's Avatar
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    (A \backslash B) \cap (A \cap B)'

    = (A \cap B') \cap (A' \cup B')

    = A \cap B' \cap (B' \cup A')

     = A \cap B'

    Where we used the following rules...

    (A \backslash B) = (A \cap B')

    (A \cap B)' = (A' \cup B')

    B' \cap (B' \cup A') = B'
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    Hello everyone

    All that Deadstar says is correct, but I think the statement
    B' \cap(B'\cup A') = B'
    (although obvious) requires further proof - it's not one of the usual Set Laws (although some people do include it as an Absorption Law).

    You prove it like this:
    B' \cap(B'\cup A') = (B' \cup \oslash)\cap(B'\cup A')
    =B'\cup(\oslash \cap A')

    =B'\cup\oslash


    =B'

    Grandad
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  5. #5
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    (A \backslash B) = (A \cap B')

    (A \cap B)' = (A' \cup B')

    B' \cap (B' \cup A') = B'[/quote]

    how to know when to apply this rule?
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  6. #6
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    Hello may
    Quote Originally Posted by may View Post
    Where we used the following rules...

    (A \backslash B) = (A \cap B')

    (A \cap B)' = (A' \cup B')

    B' \cap (B' \cup A') = B'

    how to know when to apply this rule?
    Sorry - there's no quick answer to that - just lots of practice!

    Grandad
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  7. #7
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    Quote Originally Posted by Grandad View Post
    Hello everyone

    All that Deadstar says is correct, but I think the statement
    B' \cap(B'\cup A') = B'
    (although obvious) requires further proof - it's not one of the usual Set Laws (although some people do include it as an Absorption Law).

    You prove it like this:
    B' \cap(B'\cup A') = (B' \cup \oslash)\cap(B'\cup A')
    =B'\cup(\oslash \cap A')

    =B'\cup\oslash


    =B'

    Grandad
    which formula Grandad are using to change fr

    [/SIZE][INDENT] B' \cap(B'\cup A') = (B' \cup \oslash)\cap(B'\cup A')
    [INDENT] =B'\cup(\oslash \cap A')

    =B'\cup\oslash
    [SIZE=3]
    Last edited by may; May 26th 2010 at 08:02 AM. Reason: Fixed LaTeX
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  8. #8
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    Quote Originally Posted by Grandad View Post
    Hello maySorry - there's no quick answer to that - just lots of practice!

    Grandad
    HOW to do this type of question?Must we memorise All the formula?
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  9. #9
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    Hello may
    Quote Originally Posted by may View Post
    which formula grandad are using to change fr
    B' \cap(B'\cup A') = (B' \cup \oslash)\cap(B'\cup A')

    =B'\cup(\oslash \cap A')
    This uses the Distributive Law:
    (P \cup Q)\cap (P \cup R) = P \cup (Q \cap R)
    where I have replaced P by B', Q by \oslash and R by A'.
    Quote Originally Posted by may View Post
    HOW to do this type of question?Must we memorise All the formula?
    Yes, I'm rather afraid that you do. As I said - you'll need lots of practice!

    Grandad

    PS ... and to answer your extra question, following your edit:

    \oslash \cap A' = \oslash uses what I have always known as an Identity Law, but others sometimes refer to as a Domination Law.

    See, for example, here.
    Last edited by Grandad; May 26th 2010 at 08:16 AM. Reason: Add PS
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  10. #10
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    Quote Originally Posted by Grandad View Post
    Hello mayThis uses the Distributive Law:
    (P \cup Q)\cap (P \cup R) = P \cup (Q \cap R)
    where I have replaced P by B', Q by \oslash and R by A'.

    Yes, I'm rather afraid that you do. As I said - you'll need lots of practice!

    Grandad

    PS ... and to answer your extra question, following your edit:

    \oslash \cap A' = \oslash uses what I have always known as an Identity Law, but others sometimes refer to as a Domination Law.

    See, for example, here.
    Oh,thanks Grandad at last i start to figure out how to do this type of question.thank you very much
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  11. #11
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    Quote Originally Posted by Grandad View Post
    Hello mayThis uses the Distributive Law:
    (P \cup Q)\cap (P \cup R) = P \cup (Q \cap R)
    where I have replaced P by B', Q by \oslash and R by A'.

    Yes, I'm rather afraid that you do. As I said - you'll need lots of practice!

    Grandad

    PS ... and to answer your extra question, following your edit:

    \oslash \cap A' = \oslash uses what I have always known as an Identity Law, but others sometimes refer to as a Domination Law.

    See, for example, here.
    Oh,thanks Grandad at last i start to figure out how to do this type of question.thank you very much


    erm i want to ask another question..same topic do i need to put the question in new thread or at here?
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  12. #12
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    Hello may
    Quote Originally Posted by may View Post
    Oh,thanks Grandad at last i start to figure out how to do this type of question.thank you very much


    erm i want to ask another question..same topic do i need to put the question in new thread or at here?
    Start a new thread.

    Grandad
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