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Let A, B and C be sets. Prove that

  1. Feb 23, 2012 #1
    *Sorry wrong section*
    Let A, B and C be sets.
    Prove that if A[itex]\subseteq[/itex]B[itex]\cup[/itex]C and A[itex]\cap[/itex]B=∅, then A[itex]\subseteq[/itex]C.

    My attempted solution:
    Assume A[itex]\subseteq[/itex]B[itex]\cup[/itex]C and A[itex]\cap[/itex]B=∅.
    Then [itex]\vee[/itex]x (x[itex]\in[/itex]A[itex]\rightarrow[/itex]x[itex]\in[/itex]B[itex]\cup[/itex]x[itex]\in[/itex]c).

    I'm not sure where to start and how to prove this. Any help would be greatly appreciated. Thank you.
     
    Last edited: Feb 23, 2012
  2. jcsd
  3. Feb 23, 2012 #2

    jedishrfu

    Staff: Mentor

    you almost have it.

    for all x in A and A is a subset of B U C then x is in B U C
    if x is in B U C then x is in B or x is in C

    now just describe x with respect to the A [itex]\bigcap[/itex] B = ∅
     
  4. Feb 23, 2012 #3

    Bacle2

    User Avatar
    Science Advisor

    Another perspective: you can maybe rigorize it by saying:

    a in B or a in C , but a not in B , so we must have a in C:

    Basically, as you rightly concluded, a must be in one

    of B or C, but ,by assumption/construction, a is not in B,

    so a must be in C.
     
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