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Math Help - Element-wise proof of sets

  1. #1
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    Element-wise proof of sets

    Hi, I need help on this problem:

    Let A and B be two events. Prove the following relations by the element-wise method.

     (A-AB) \cup B = A \cup B

    When I say element-wise method, the idea is to show that the events on both sides of the equation are formed of the same sample point. We have to prove:

     (1) (A-AB) \cup B = A \cup B

    and the reverse

     (2) A \cup B = (A-AB) \cup B

    Any ideas?
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  2. #2
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    \begin{gathered}<br />
  x \in \left( {A \cup B} \right) \Rightarrow \quad x \in A \vee x \in B \hfill \\<br />
  x \notin B \Rightarrow \quad x \in \left( {A\backslash B} \right) \Rightarrow \quad x \in \left( {A\backslash B} \right) \cup B \hfill \\<br />
  x \in B \Rightarrow \quad x \in \left( {A\backslash B} \right) \cup B \hfill \\ <br />
\end{gathered}

    Now you do the other half.
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