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Math Help - symmetric difference of sets

  1. #1
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    symmetric difference of sets

    prove that for sets A,B,C,D
    A \Delta B = C \Delta D if and only if A \Delta C = B \Delta D
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by santiagos11 View Post
    prove that for sets A,B,C,D
    A \Delta B = C \Delta D if and only if A \Delta C = B \Delta D
    What have you tried? Although there are many I think that the definition of symmetric difference best suited to this case is A\Delta B=\left(A\cup B\right)-\left(A\cap B\right).
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  3. #3
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    There is also a different way to solve this that sometimes works very well with symmetric difference.

    Define the function in by in(x,A)=1 if x\in A and in(x,A)=0 if x\notin A. Here A is any set and x is any object.

    Also, define the function \oplus:\{0,1\}\times\{0,1\}\to\{0,1\} as follows:
    0\oplus0=0
    0\oplus1=1
    1\oplus0=1
    1\oplus1=0
    (If we associate 1 with True and 0 with False, then x\oplus y=\neg(x\leftrightarrow y). Also, \oplus is addition modulo 2.) One can check that \oplus is commutative and associative.

    Show that in(x,A\triangle B)=in(x,A)\oplus in(x,B) for all A, B, and x.

    The part above was generic; it applies to any problem with symmetric difference and has to be done only once.

    Now, from this problem's assumption and the fact above we have in(x,A)\oplus in(x,B)=in(x,C)\oplus in(x,D) for all x. We can add (using \oplus) an expression in(x,E) to both sides where E is A, B, C, or D. E.g., if we add in(x,B) to both sides, we get in(x,A)= in(x, C)\oplus in(x,D)\oplus in(x,B). From here it is easy to show the desired equation.
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  4. #4
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    Hello santiagos11
    Quote Originally Posted by santiagos11 View Post
    prove that for sets A,B,C,D
    A \Delta B = C \Delta D if and only if A \Delta C = B \Delta D
    I think this is easiest to prove by re-writing it as the equivalent statement using logical propositions. In other words:
    Prove that if a, b,c, d are logical propositions, then a \oplus b \equiv c \oplus d if and only if a \oplus c \equiv b \oplus d, where \oplus denotes 'exclusive or'.
    This is quite straightforward if you use a Truth Table for each of the compound propositions and show that each table produces an identical output.

    The attached diagram shows the truth table for a \oplus b \equiv c \oplus d. It is a simple matter to show that the table for a \oplus c \equiv b \oplus d produces the same result.

    Grandad
    Attached Thumbnails Attached Thumbnails symmetric difference of sets-untitled.jpg  
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