Results 1 to 3 of 3

Math Help - Proof: A-B⊆C if and only if A-C⊆B.

  1. #1
    Newbie
    Joined
    Apr 2012
    From
    Greenville, SC
    Posts
    4

    Proof: A-B⊆C if and only if A-C⊆B.

    Having trouble figuring this proof out. Can anyone help me?

    Prove A-B⊆C if and only if A-C⊆B.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,527
    Thanks
    773

    Re: Proof: A-B⊆C if and only if A-C⊆B.

    Note that X ⊆ Y is equivalent to X ∩ Y' = ∅ where Y' is the complement of Y and ∅ is the empty set. And, as you already know, X - Y = X ∩ Y'. Using these facts, rewrite both sides.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,541
    Thanks
    1394

    Re: Proof: A-B⊆C if and only if A-C⊆B.

    You can also do this directly from the definitions. To prove " X\subseteq Y" show that "if x is in X then it is in Y".

    Here, we want to prove A- C\subseteq B so start "if x is in A- C" and try to conclude that x is in B. If x is in A- C then it is in A but not in C. But we know that A- B\subseteq C so every member of A except those that are also in B. Since x is in A but NOT in C, it is in B.

    To prove the other way "if A- C\subseteq B then A- B\sdubseteq C do the opposite "if x is in A- B" and show it must be in C.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: September 19th 2011, 12:42 PM
  2. Replies: 5
    Last Post: October 19th 2010, 10:50 AM
  3. [SOLVED] direct proof and proof by contradiction
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: February 27th 2010, 10:07 PM
  4. Proof with algebra, and proof by induction (problems)
    Posted in the Discrete Math Forum
    Replies: 8
    Last Post: June 8th 2008, 01:20 PM
  5. proof that the proof that .999_ = 1 is not a proof (version)
    Posted in the Advanced Applied Math Forum
    Replies: 4
    Last Post: April 14th 2008, 04:07 PM

Search Tags


/mathhelpforum @mathhelpforum