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Set Equivalencies and stuff

  1. Jul 13, 2011 #1
    1. The problem statement, all variables and given/known data
    Alright so I was trying to solve this using logical equivalences:

    Fill in the blanks to make true identities:
    [itex] C \backslash ( A \Delta B) = (A \cap C) \Delta [/itex] ______

    I made it to the end where I stated that the missing part was (C\B), but I'm not sure if my last step was justified

    2. Relevant equations
    Equivalences


    3. The attempt at a solution [\b]

    I'll skip most of the steps (there were about 9) because I suck at latex but the last few are (working from the left side):

    [itex] [ ( x \in C \wedge x \in A) \wedge (x \notin A \vee x \in B) ] \vee [ (x \in C \wedge x \notin B) \wedge (x \notin A \vee x \in B) ] \\
    [ (C \cap A) \cap (B \cup (x \notin A) ] \cap [ (C \backslash B) \cap (B \cup (x \notin A) ] \\
    C \backslash (A \Delta B) = (A \cap C) \Delta (C \backslash B) [/itex]

    So in the 2nd to last step, I dropped [tex](B \cup (x \notin A)[/tex] from both sides of the union because of the definition of symmetric difference which says that they would be dropped even if I left them in. Is this correctly justified?
     
    Last edited: Jul 13, 2011
  2. jcsd
  3. Jul 13, 2011 #2
    Screw latex, here's a scan of my work

    anyway, in the 2nd to last step I dropped (B union ...) out of both sides due to the fact that they would get dropped anyway when symmetric difference was thrown in, is this justified?
     

    Attached Files:

  4. Jul 14, 2011 #3
    Okay so I rewrote [itex] [ ( B \cup ( x \notin A ) ] [/itex] because I realized that notation doesn't make any sense (which I originally knew but wasn't sure how to express the -xEA part in set notation) and came up with [itex] [ B \cup ( C \backslash A ) ] [/itex] which I believe is a way of representing that in this context (where the only values we are talking about are represented in sets A B and C)

    Is this rewrite correct or is there some way of representing -xEA using set notation that I don't know about?

    Anyway, after plugging that in I realized that the 2nd to last line reads:
    [itex] [ ( C \cap A ) \cap ( B \cup ( C \backslash A ) ] \cup [ ( C \backslash B ) \cap ( B \cup ( C \backslash A ) ] [/itex]
    And because of the first part which reads [itex] [ ( C \cap A ) ][/itex], having [itex] [ (C \cap A ) \cap ( C \backslash A ) ] [/itex] is a contradiction, and because of that I can drop [itex] ( C \backslash A ) [/itex] from the equation on both sides.

    Then, again because of the definition of symmetric difference (and because it's also a contradiction) I drop the [itex] \cap B[/itex] from both sides because it would be removed regardless.

    Is this correct?
     
    Last edited: Jul 14, 2011
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