Intersection of Event with Complement Can't Happen

Theorem

Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.

Let $A \in \Sigma$ be an events of $\EE$, so that $A \subseteq \Omega$.

Then:

$A \cap \overline A = \O$

where $\overline A$ is the complementary event to $A$.

That is:

$A \cap \overline A$ is an impossibility

or:

$\map \Pr {A \cap \overline A} = 0$

Proof

By definition:

$A \subseteq \Omega$

and:

$\overline A = \relcomp \Omega A$

From Intersection with Relative Complement is Empty:

$A \cap \overline A = \O$

We then have from Probability of Empty Event is Zero that:

$\map \Pr \Omega = 0$

The result follows by definition of impossible event.

$\blacksquare$

Also see

• Union of Event with Complement is Certainty

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): complementary (for probability)