Correlation of Indicator Variables

Question

Show that for indicator random variables $I_A$ and $I_B$ of Events $A$ and $B$:

$Corr(I_A, I_B) = Corr(I_A^c, I_B^c) = -Corr(I_A, I_B^c) = -Corr(I_A^c, I_B)$

Deduce that if $A$ and $B$ are positively dependent, then so are $A^c$ and $B^c$, but $A$ and $B^c$ are negatively dependent, as are $A^c$ and $B$.

Not sure how to show this.

Answer 1

Note a basic rule of indicators, $I_A=1-I_\bar{A}$.

It may also help to refer to problem 6.4.16 where it asks to prove that $$ Corr(aX+b,cY+d) = \left\{ \begin{array}{ll} Corr(X,Y) & \quad \text{if $a$ and $c$ have the same sign} \\ -Corr(X,Y) & \quad \text{if $a$ and $c$ have opposite signs} \end{array} \right. $$