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Complement Set

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

\begin{displaymath}
E' \equiv \{F: F\in S, F\notin E\}.
\end{displaymath} (1)

If $E = S$, 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)




© 1996-9 Eric W. Weisstein
1999-05-26