# Element-wise proof of sets

• Oct 11th 2008, 02:58 PM
MagicS06
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?
• Oct 11th 2008, 03:37 PM
Plato
$\begin{gathered}
x \in \left( {A \cup B} \right) \Rightarrow \quad x \in A \vee x \in B \hfill \\
x \notin B \Rightarrow \quad x \in \left( {A\backslash B} \right) \Rightarrow \quad x \in \left( {A\backslash B} \right) \cup B \hfill \\
x \in B \Rightarrow \quad x \in \left( {A\backslash B} \right) \cup B \hfill \\
\end{gathered}$

Now you do the other half.