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Proving a set identity

  1. Dec 1, 2016 #1
    1. The problem statement, all variables and given/known data
    The problem is to prove that ##A \cup (B - A) = \varnothing##

    2. Relevant equations


    3. The attempt at a solution
    The solution in the textbook is that
    ##A \cup (B-A) = \{x~ |~ x \in A \land (x \in B \land x \not\in A) \} = \{x~ |~ x \in A \land x \not\in A \land x \in B \} = \{x~ | ~ F\} = \varnothing##. I am just confused as to why ##\{x~ | ~ F\} = \varnothing##. Why is that logically a consequence?
     
  2. jcsd
  3. Dec 1, 2016 #2

    fresh_42

    Staff: Mentor

    Well, first of all, it should be an intersection ##\cap## (and), not a union ##\cup## (or).
    Then ##\{x\,\vert \,F\}## is the set of all ##x##, which satisfy ##"false"##. But ##"false"## is never satisfied, thus ##\emptyset##.
     
  4. Dec 1, 2016 #3

    Mark44

    Staff: Mentor

    To expand on what fresh_42 said, the expression on the right is the definition of ##A \cap (B - A)##; i.e., the intersection of A and B - A, not the union of these two sets.
     
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