Complement Set

Given a set with a subset , the complement of is defined as

$\begin{displaymath} E' \equiv \{F: F\in S, F\notin E\}. \end{displaymath}$

(1)

, then

$\begin{displaymath} E' \equiv S' = \emptyset, \end{displaymath}$

(2)

where $\emptyset$ is the Empty Set. Given a single Set, the second Probability Axiom gives

$\begin{displaymath} 1 = P(S) = P(E\cup E'). \end{displaymath}$

(3)

Using the fact that $E\cap E' = \emptyset$ ,

$\begin{displaymath} 1 = P(E)+P(E') \end{displaymath}$

(4)

$\begin{displaymath} P(E') = 1-P(E). \end{displaymath}$

(5)

This demonstrates that

$\begin{displaymath} P(S') = P(\emptyset) = 1-P(S) = 1-1 = 0. \end{displaymath}$

(6)

Given two Sets,

$\displaystyle P(E\cap F')$	$\textstyle =$	$\displaystyle P(E)-P(E\cap F)$	(7)
$\displaystyle P(E'\cap F')$	$\textstyle =$	$\displaystyle 1-P(E)-P(F)+P(E\cap F).$	(8)