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Does negating a set change it symbolically?

  1. Feb 25, 2016 #1
    So I have to prove "If (AxB)∩(BxA) ≠ ∅, then (A∩B) ≠ ∅." I wanted to prove by changing it's form.
    P = (AxB)∩(BxA) ≠ ∅ and Q = (A∩B) ≠ ∅ . The conditional statement is P implies Q and the new statement is not P or Q .
    P → Q = ¬ P∨Q They are equivalent.
    But how do I negate P?
    Would it be (AxB)∩(BxA) = ∅ instead of (AxB)∩(BxA) ≠ ∅ or does the left side of the equal sign also change?

    Also, is this the easiest way of proving this theorem? Is there any easier way? Or should I negate the entire thing and it comes out false, then the original statement is true...
     
  2. jcsd
  3. Feb 25, 2016 #2

    Samy_A

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    There is an easier way.
    If (AxB)∩(BxA) ≠ ∅,then (AxB)∩(BxA) contains at least one element. Work with that element: find out
    how you can represent an element of (AxB)∩(BxA).
     
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