Show that $\max\{\textbf{P}((A\cup B)^{c}),\textbf{P}(A\cap B),\textbf{P}(A\triangle B)\}\geq\frac{4}{9}$ Let $A$ and $B$ be independent events. Show that
\begin{align*}
\max\{\textbf{P}((A\cup B)^{c}),\textbf{P}(A\cap B),\textbf{P}(A\triangle B)\}\geq\frac{4}{9}
\end{align*}
MY ATTEMPT
Since $\textbf{P}(A\cap B) = \textbf{P}(A)\textbf{P}(B)$, we have
\begin{align*}
\textbf{P}((A\cup B)^{c}) & = 1 - \textbf{P}(A\cup B) = 1 - \textbf{P}(A) - \textbf{P}(B) + \textbf{P}(A\cap B)\\
& = 1 - \textbf{P}(A) - \textbf{P}(B) + \textbf{P}(A)\textbf{P}(B)\\
& = (1 - \textbf{P}(A)) - \textbf{P}(B)(1 - \textbf{P}(A))\\
& = (1 - \textbf{P}(A))(1-\textbf{P}(B)) = \textbf{P}(A^{c})\textbf{P}(B^{c})
\end{align*}
Analagously, we get
\begin{align*}
\textbf{P}(A\triangle B) & = \textbf{P}(A) + \textbf{P}(B) - 2\textbf{P}(A\cap B) = \textbf{P}(A) + \textbf{P}(B) - 2\textbf{P}(A)\textbf{P}(B)\\\\
& = \textbf{P}(A)(1 - \textbf{P}(B)) + \textbf{P}(B)(1-\textbf{P}(A)) = \textbf{P}(A)\textbf{P}(B^{c}) + \textbf{P}(A^{c})\textbf{P}(B)
\end{align*}
However, I do not know how to proceed from here. Am I on the right track? Could someone complete the proof? Any help is appreciated. Thanks in advance.
 A: As in the comment by Lord Shark The Unknown you can first set
$$ P(A) = p$$
and
$$P(B) = q.$$
Then the thesis becomes the following.

Show that the inequalities
  \begin{eqnarray}
pq &<& \frac{4}{9}\\
(1-p)(1-q) &<& \frac{4}{9}\\
p(1-q) + q(1-p) &<& \frac{4}{9}
\end{eqnarray}
  cannot hold simultaneously.

Performing the following change of variables
\begin{equation}
\begin{cases}
p = \frac{1}{2}(Q+P+1)\\
q = \frac{1}{2}(Q-P+1)
\end{cases}
\end{equation}
leads to the corresponding inequalities in terms of $P$ and $Q$, i.e.
\begin{eqnarray}
(Q+1)^2 -P^2 &<& \frac{16}{9}\tag{1}\label{uno}\\
(Q-1)^2 -P^2 &<& \frac{16}{9}\tag{2}\label{due}\\
Q^2-P^2>\frac{1}{9}\tag{3}\label{tre}
\end{eqnarray}
which are areas delimited by hyperbolae (in red, green, and black respectively in the Figure). The red dashed area corresponds to the set of points $(P,Q)$ satisfying conditions \eqref{uno} and \eqref{due}. The black dashed areas mark te set of points that satisfy condition \eqref{tre}. Therefore the three inequalities cannot be all true. And the thesis follows.

