Results 1 to 4 of 4

Math Help - Set Theory Proof

  1. #1
    Newbie
    Joined
    Jul 2006
    Posts
    12

    Set Theory Proof

    Just wondering if someone could help me with this proof from a practice exam I'm studying for.

    Let A and B be sets. Prove or disprove:
    (B-A) union (A-B) = (A union B) - (A intersection B)

    Thanks for any help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,083
    Thanks
    375
    Awards
    1
    Quote Originally Posted by OntarioStud View Post
    Just wondering if someone could help me with this proof from a practice exam I'm studying for.

    Let A and B be sets. Prove or disprove:
    (B-A) union (A-B) = (A union B) - (A intersection B)

    Thanks for any help.
    There are probably some set rules to break this down, but look at it this way: there are only 4 possibilities that you need to check.
    1) A and B are disjoint.
    2) A and B intersect, but do not contain one or the other.
    3) A is a subset of B.
    4) B is a subset of A.

    (I suppose you ought to include the possibilities were either A or B or both are the empty set, but these should be easy.)

    -Dan
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,803
    Thanks
    1692
    Awards
    1
    (A-B)U(B-A)
    =(A^B)U(B^A)
    =(AUB)^(BUB)^(AUB)^(BUA)
    =(AUB)^(BUA) [(AUA)^(BUB) is space.]
    =(AUB)^(B^A)
    =(AUB)-(A^B)

    Too bad TeX is down.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,814
    Thanks
    703
    Hello, OntarioStud!

    I will use A' for "complement of A" and 0 for the empty set.


    Let A and B be sets.
    Prove or disprove: .(B - A) U (A - B) = (A U B) - (A ∩ B)

    Defintion of set subtraction: .A - B .= .A ∩ B'

    Axiom #1: .A ∩ A' = 0

    Axiom #2: .A U 0 = A

    Distributive Laws: .A U (B ∩ C) = (A U B) ∩ (A U C)
    . . . . . . . . . . . . . .A ∩ (B U C) = (A ∩ B) U (A ∩ C)


    The right side is: .(A U B) - (A ∩ B)

    . . . . . . . . . . .= .(A U B) ∩ (A ∩ B)' . Def. of Subtraction

    . . . . . . . . . . .= .(A U B) ∩ (A' U B') . DeMorgan's Law

    . . . . . . . . . . .= .[(A U B) ∩ A'] U [(A U B) ∩ B'] . Distr.Law

    . . . . . . . . . . .= .[(A ∩ A') U (B ∩ A')] U [(A ∩ B') U (B ∩ B')] . Distr.Law

    . . . . . . . . . . .= .[0 U (B ∩ A')] U [(A ∩ B') U 0] . Axiom #1

    . . . . . . . . . . .= .(B ∩ A') U (A ∩ B') . Axiom #2

    . . . . . . . . . . .= .(B - A) U (A - B) . Def. of Subtraction

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help with set theory proof
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: March 8th 2011, 10:25 AM
  2. Replies: 5
    Last Post: October 19th 2010, 10:50 AM
  3. Set theory proof?
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: April 10th 2010, 12:27 PM
  4. set theory proof
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: February 25th 2010, 06:11 PM
  5. Set Theory Proof
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 2nd 2010, 03:40 AM

Search Tags


/mathhelpforum @mathhelpforum