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Set Theory Identities

  1. Jun 15, 2010 #1
    1. The problem statement, all variables and given/known data

    Can you conclude that A = B if A, B, and C are sets such that

    A [tex]\cup[/tex] C = B [tex]\cup[/tex] C and A [tex]\cap[/tex] C = B [tex]\cap[/tex] C


    2. Relevant equations

    The above is part c of a problem. The problems a and b are as follows

    A) A [tex]\cup[/tex] C = B [tex]\cup[/tex] C

    My answer: I gave a counter example such that A = {a, b, c}, B = {c, d, e}
    and C = {a, b, c, d, e}, thus A [tex]\cup[/tex] C = C = B [tex]\cup[/tex] C
    but A [tex]\neq[/tex] B

    B) A [tex]\cap[/tex] C = B [tex]\cap[/tex] C

    My answer: I gave the counter example where A = {a, b, c}, B = {c, d, e}, C = {c}
    So, A [tex]\cap[/tex] C = C = B [tex]\cap[/tex] C but A [tex]\neq[/tex] B

    3. The attempt at a solution

    Ok for this part c I could not think of a counter example. I believe they want me to use set identities. I'm honestly not sure where to begin but Ill tell you what I have in mind so far.

    If A [tex]\cup[/tex] C = B [tex]\cup[/tex] C, this implies that (A [tex]\cup[/tex] C) [tex]\subseteq[/tex] (B [tex]\cup[/tex] C), and (B [tex]\cup[/tex] C) [tex]\subseteq[/tex]
    (A [tex]\cup[/tex] C)

    So, (A [tex]\cup[/tex] C) [tex]\subseteq[/tex] (B [tex]\cup[/tex] C)

    Same goes for (A [tex]\cap[/tex] C) [tex]\subseteq[/tex] ( B [tex]\cap[/tex] C),

    In order to prove A = B I need to prove A [tex]\subseteq[/tex] B and B [tex]\subseteq[/tex] A.

    So I have these premises and a conclusion, but Im honestly not sure how to set this up. I'm pretty sure I need to use set identities.. If anyone has any advice to get me moving here I'd greatly appreciate it, thanks!
     
  2. jcsd
  3. Jun 15, 2010 #2

    Hurkyl

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    Gold Member

    I don't think it's too hard to prove this by looking at elements in combination with algebraic identities, rather than a purely algebraic proof, if you are inclined to do so.


    Rather than trying to prove that A=B, you may find it easier constructing a set that measures how different A and B are, and then proving something about that.


    What identities are you considering using? Rewriting equality in terms of subsets is a place to start, but you don't seem to have invoked any properties of union and intersection yet.

    I imagine you're probably using a list like the one here, along with the ones in the next section relating meet and join to ordering.

    (In your case, the set-theoretic symbols [itex]\cap, \cup, \subseteq[/itex] correspond to the lattice algebra symbols [itex]\wedge, \vee, \leq[/itex])

    FYI, that list of identities is not enough. You need these as well.

    You may find identities relating to complementation useful too.
     
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