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Set Theory Proof

  1. Apr 4, 2009 #1
    1. The problem statement, all variables and given/known data
    For all sets A and B, if A [tex]\subseteq[/tex] B then Bc [tex]\subseteq[/tex] Ac.


    2. Relevant equations



    3. The attempt at a solution
    Proof: Suppose A and B are sets and A [tex]\subseteq[/tex] B.
    Let x [tex]\in[/tex] Bc
    By definition of complement, if x [tex]\in[/tex] Bc then x [tex]\notin[/tex] B
    Since x [tex]\notin[/tex] B, x [tex]\notin[/tex] A
    Since x [tex]\notin[/tex] A, x [tex]\in[/tex] Ac by definition of complement
    Therefore if A [tex]\subseteq[/tex] B then Bc [tex]\subseteq[/tex] Ac.

    I just want to make sure that this proof is correct and that there are no mistakes. Thanks!
     
  2. jcsd
  3. Apr 4, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    It's fine. You might want to enhance it's proofiness by stating the reason why x not in B implies x not in A as you gave a reason for the other lines.
     
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