1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Staff Emeritus
    Science Advisor
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook