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  • negative sets and commutative and associative addition and subtraction

    From =?UTF-8?Q?Hagen_Schwa=C3=9F?=@21:1/5 to All on Sat Jul 25 09:28:02 2020
    Please read up

    https://groups.google.com/forum/#!topic/sci.math/8sY4Mcgkm1g

    if it could help for databases.

    This is a quick start:

    Set U, e.g. R^2

    Set U* set of all subsets of U including U and {}

    Definitions:

    A,B in U*

    AnB {x:x in A and x in B} intersection
    AuB {x:x in A or X in B} union
    A\B {x:x in A and x not in B} without

    Note that traditional set operations bind harder than definitions: (AuB)+(AnB). Further -negation binds hardest: -AuB=(-A)uB.

    We are now defining algebraic compatible +/- operators:

    A+B=AnB+AuB
    A-B=A\B-B\A

    A+{}=A
    {}=-{}

    Proof:

    A-B=A+(-B)=An-B+Au-B=A\B-B\A
    we choose AnB=A\B and AuB=-(B\A)

    -A-B=-(A+B)=-Au-B+-An-B=-(AnB)-(AuB)
    we choose -(AnB)=-Au-B and -(AuB)=-An-B

    it holds:

    -A\-B=B\A

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  • From =?UTF-8?Q?Hagen_Schwa=C3=9F?=@21:1/5 to All on Sat Jul 25 09:38:32 2020
    Am Samstag, 25. Juli 2020 18:28:03 UTC+2 schrieb Hagen Schwaß:
    Please read up

    https://groups.google.com/forum/#!topic/sci.math/8sY4Mcgkm1g

    if it could help for databases.

    This is a quick start:

    Set U, e.g. R^2

    Set U* set of all subsets of U including U and {}

    Definitions:

    A,B in U*

    AnB {x:x in A and x in B} intersection
    AuB {x:x in A or X in B} union
    A\B {x:x in A and x not in B} without

    Note that traditional set operations bind harder than definitions: (AuB)+(AnB). Further -negation binds hardest: -AuB=(-A)uB.

    We are now defining algebraic compatible +/- operators:

    A+B=AnB+AuB
    A-B=A\B-B\A

    A+{}=A
    {}=-{}

    Proof:

    A-B=A+(-B)=An-B+Au-B=A\B-B\A
    we choose AnB=A\B and AuB=-(B\A)

    -A-B=-(A+B)=-Au-B+-An-B=-(AnB)-(AuB)
    we choose -(AnB)=-Au-B and -(AuB)=-An-B

    it holds:

    -A\-B=B\A

    We proof associativity and commutativity by using n*n*(n+1)/2 atomic subsets. Please read up

    https://groups.google.com/forum/#!topic/de.sci.mathematik/2ndfhuWhwzQ

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