For any events $A$ and $B$, prove that $A \cup B = (AB^c)\cup(AB)\cup(A^cB)$.

elementary-set-theorylogicprobability

For any events $A$ and $B$, prove that $A \cup B = (AB^c)\cup(AB)\cup(A^cB)$. This result is used to prove that $P(A \cup B) = P(A) + P(B) – P(AB)$ for any events $A$ and $B$.

The result is apparent when one draws a Venn diagram, but how can it be proven mathematically?

EDIT:

I suppose this is a more general form of: if $\omega \in \Omega$, then either $\omega \in A$ or $\omega \in A^c$. Going a step further, $A = AB^c \bigcup AB$ because if $\omega \in A$, it is also either in $B$ or $B^c$ . Are statements such as these obvious, or are they true due to "insert theorem name here?"

Best Answer

You can just use:

Adjacency

$AB \cup AB^C=A$

and you also have:

Idempotence

$A \cup A =A$

So:

$AB^C \cup AB \cup A^CB \overset{Idempotence}= AB^C \cup AB \cup AB \cup A^CB \overset{Adjacency \ x \ 2}= A \cup B$

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