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Abstract Algebra set theory

  1. Feb 23, 2012 #1
    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.
     
  2. jcsd
  3. Feb 23, 2012 #2

    jedishrfu

    Staff: Mentor

    ....disregard earlier post...
     
  4. Feb 23, 2012 #3

    jbunniii

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    Suppose [itex]x \in A[/itex]. The goal is to show that this implies [itex]x \in C[/itex].

    Since [itex]x \in A[/itex] and [itex]A \subset B \cup C[/itex], it follows that [itex]x \in B[/itex] or [itex]x \in C[/itex]. Can you exclude one of these possibilities?
     
  5. Feb 23, 2012 #4
    Since A[itex]\bigcap[/itex]B=∅, x[itex]\in[/itex]A or x[itex]\in[/itex]B.
    Thus x[itex]\in[/itex]A, x[itex]\notin[/itex]B and x[itex]\in[/itex]C.
    Therefor A[itex]\subseteq[/itex]C.

    Is that a good way to show how to exclude the possibility of x[itex]\in[/itex]B?
     
  6. Feb 23, 2012 #5

    jbunniii

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    You have the right idea, but the wording is a little unclear. The following is not true: "Since A[itex]\bigcap[/itex]B=∅, x[itex]\in[/itex]A or x[itex]\in[/itex]B."

    How about the following:

    Suppose [itex]x \in A[/itex]. The goal is to show that this implies [itex]x \in C[/itex].

    Since [itex]x \in A[/itex] and [itex]A \subset B \cup C[/itex], it follows that [itex]x \in B[/itex] or [itex]x \in C[/itex]. However, [itex]x[/itex] cannot be in [itex]B[/itex], because if it were, then we would have [itex]x \in A \cap B = \emptyset[/itex], which is impossible. Therefore...
     
  7. Feb 23, 2012 #6
    Ok I think I get it tell me if I worded this correctly:

    Suppose [itex]x \in A[/itex] and [itex]A \cap B = \emptyset[/itex]
    Since [itex]x \in A[/itex] and [itex]A \subset B \cup C[/itex], this means that [itex]x \in B[/itex] or [itex]x \in C[/itex]. Consequently, [itex]x[/itex] cannot be in [itex]B[/itex], because if it were, then we would have [itex]x \in A \cap B = \emptyset[/itex], which is impossible. Therefore [itex]x \in C[/itex]. Thus [itex]x \in A[/itex] implies [itex]x \in C[/itex] so it follows that [itex]A \subseteq C[/itex]
     
  8. Feb 23, 2012 #7

    jbunniii

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    Looks good to me.
     
  9. Feb 23, 2012 #8
    ok thank you for your help
     
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