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Discrete Math Set Theory Question

  1. Sep 15, 2012 #1
    Let A, B, and C be sets. Show that

    a) (A-B) - C [itex]\subseteq[/itex] A - C
    b) (B-A) [itex]\cup[/itex] (C-A) = (B [itex]\cup[/itex] C) - A

    I am using variable x to represent an element.

    Part A)

    I rewrote (A-B) - C as (x[itex]\in[/itex]A ^ x[itex]\notin[/itex]B) - C

    I think this could be rewritten as
    (x[itex]\in[/itex]A ^ x[itex]\notin[/itex]B) ^ x[itex]\notin[/itex] C

    A-C can be rewritten as (x [itex]\in[/itex] A ^ x [itex]\notin[/itex] C)

    The original statement can be rewritten as

    x[itex]\in[/itex]A[itex]\cap[/itex]~B[itex]\cap[/itex]~C [itex]\subseteq[/itex] x[itex]\in[/itex]A[itex]\cap[/itex]~C

    where ~ represents negation.

    However, for the LHS to be a subset of the RHS, all elements of the LHS should be an element of RHS but since the LHS has ~B, I don't think that it is a subset?

    I have no idea how to show part B so any help would be great.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 15, 2012 #2

    jedishrfu

    Staff: Mentor

    when you draw a venn diagram you'll see that set A-C contains the intersection A,B and C but that A-B-C does not. perhaps you use that in your proof. I don't think venn diagrams can be used in proofs but they do help visualize the situation.
     
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