Results 1 to 3 of 3

Math Help - Addition and multiplication with Sets

  1. #1
    Newbie
    Joined
    Nov 2007
    Posts
    10

    Addition and multiplication with Sets

    Problem:
    Let S be a set. For any two subsets of S we define
    A + B = (A-B) U (B-A)
    A*B = A \cap B
    Also I know that A + B = B +A
    A + {} = A
    A * A = A
    A + A = {}
    Prove that:
    a) A + (B + C) = (A+B)+C
    b) If A + B = A + C then B = C
    c) A * (B + C) = A*B + A*C

    a) let x \epsilon A + (B+C)
    either x \epsilon A or x \epsilon B or x \epsilon C but x belongs to only one of these sets otherwise x couldn't be in A + (B+C)
    if x \epsilon A then x \epsilon (A+B)+C
    if x \epsilon B then x \epsilon (B + A)+C = (A+B)+C
    if x \epsilon C then x doesn't belong to (A+B) and x \epsilon (A+B)+C
    so A + (B+C) is a subset of (A+B)+C
    The same agument works for using x \epsilon (A+B)+C and showing that (A+B)+C is a subset of A+(B+C) thus A+(B+C) = (A+B)+C

    b)let x \epsilon B
    if x \epsilon A+B then x \epsilon A+C and x doesn't belong in A so x \epsilon C thus B is a subset of C.
    if x doesn't belong to A+B then x doesn't belong to A+C and \epsilon A so x \epsilonC so B is a subset of C. The same agument works starting with x \epsilon C so since B is a subset of C and C is a subset of B then B=C.

    c) I'm not really sure where to start on this one.

    Are the two proof's reasonable?
    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,395
    Thanks
    1481
    Awards
    1
    The way ‘+’ is defined depends upon on the idea of “symmetric difference” of two sets.
    It is well know that using that definition an algebra of set can be defined.
    You need to look up the concept of “symmetric difference”.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2007
    Posts
    10
    The problem defines + as A + B = (A-B) U (B-A) which is consistent with one of the definition's of "symmetric difference" I found. For the problems I just thought of it as exlusive or.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. addition progression in multiplication
    Posted in the Algebra Forum
    Replies: 3
    Last Post: September 27th 2011, 11:10 AM
  2. Replies: 1
    Last Post: June 21st 2011, 06:41 AM
  3. lim sup and addition, multiplication
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 3rd 2009, 04:05 PM
  4. Multiplication and Addition rules
    Posted in the Statistics Forum
    Replies: 1
    Last Post: May 1st 2008, 10:17 AM
  5. Equations - both addition and multiplication
    Posted in the Algebra Forum
    Replies: 4
    Last Post: February 26th 2006, 08:38 PM

Search Tags


/mathhelpforum @mathhelpforum