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Proofing an uncertain theorem to verify it's truth?

  1. Sep 3, 2014 #1
    1. The problem statement, all variables and given/known data
    Suppose R is a relation from A to B and S and T are relations from
    B to C.
    Must it be true that (S \ T ) ◦ R ⊆ (S ◦ R) \ (T ◦ R)?

    3. The attempt at a solution
    I assume that (S \ T ) ◦ R ⊆ (S ◦ R) \ (T ◦ R) is true and attempt to prove it (if I run into a contradiction I know that it is false)

    Part 1
    let [itex](a,c) \in (S\T ) ◦ R[/itex] then, we can choose a [itex]b \in B[/itex] such that,

    [itex]\exists b \in B\ ((a,b) \in R\land(b,c) \in (S\T))[/itex]

    Therefore,
    [itex](a,b) \in R[/itex]
    [itex](b,c) \in S[/itex]
    [itex](b,c) \notin T[/itex]

    But this means that [itex](a,c) \in (S\circ R)[/itex]

    Part 2
    To prove that [itex](a,c) \notin (T\circ R)[/itex] we employ proof by contradiction:

    Suppose [itex](a,c) \in (T\circ R)[/itex] Then we instantiate a [itex]b \in B[/itex], say, b' such that:
    [itex](a,b') \in R[/itex]
    [itex](b',c) \in T[/itex]
    -------------

    I've hit a dead-end here and I'm not sure how to progress any further. As an exercise I wanted to refrain from plugging-and-chugging random scenarios into the theorem to prove it's veracity and decided instead to reach a contradiction within a contradiction to prove that the theorem is in fact, false.

    However, I haven't (or at least I don't believe) that I have reached a contradiction in the proof, I'm just at a dead end (or at least that's how it seems).

    Any insight would be much appreciated.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 3, 2014 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    What does "R\S" mean? "Relations" are sets of pairs so I would assume that "R\S" is the set operation "all members of R that are not in S" but that can't be true here because NO member of R is in S.
     
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