A really quick question about complements and symmetric difference Is $ℙ(^△^) $ the same as $ℙ(△)$?
My inclination is no, as when I use substitution giving a number it gives two different numbers, am I correct in this assumption that they aren't the same?
 A: A really quick and easy way to resolve questions about set operations is to compute with indicator functions with values in $\mathbb{Z}/2\mathbb {Z},$ which is the set $\{0,1\}$ with addition and multiplication modulo $2.$  The only rule you need to know that isn't determined by the definitions of $0$ (the additive unit) and $1$ (the multiplicative unit) is $1+1=0.$
The indicator function of a set $A,$ written $\mathcal{I}_Z,$ takes on the value $1$ at all elements of $A$ and otherwise takes on the value $0.$  A set determines its indicator and vice versa, because when $f$ is any function taking values in $\mathbb{Z}/2\mathbb{Z},$ $f = \mathcal{I}_A$ where $A = \{a\mid f(a)=1\}.$
Indicator functions are added and multiplied like any other functions: by adding and multiplying their values at each element of their domains.
When the sets are all elements of a common "universe," let "$1$" be the indicator of the universe.
In these terms, here are how set operations correspond to indicator operations.
$$\begin{array}
&\text{Name}&\text{Sets} & \text{Indicators}\\\hline
\text{Intersection}&A\cap B & \mathcal{I}_A\mathcal{I}_B\\
\text{Symmetric difference}&A\Delta B &\mathcal{I}_A + \mathcal{I}_B\\
\text{Complement}&A^c & 1+\mathcal{I}_A\\
\text{Union}& A\cup B & \mathcal{I}_A + \mathcal{I}_B + \mathcal{I}_A\mathcal{I}_B\\
\text{Difference}& A\setminus B & \mathcal{I}_A(1 + \mathcal{I}_B)
\end{array}$$
Thus, because $1+1=0$ in $\mathbb{Z}/2\mathbb Z,$ to $A^c\Delta B^c$ corresponds the indicator
$$(1 + \mathcal{I}_A) + (1 + \mathcal{I}_B) = 1 + 1 + \mathcal{I}_A+\mathcal{I}_B = \mathcal{I}_A+\mathcal{I}_B,$$
which is the indicator of $A\Delta B.$  Therefore $A\Delta B = A^c\Delta B^c.$ Since these sets are always equal, their probabilities must also be equal, no matter what the probability distribution might be.  If you computed different values then you made a mistake somewhere.
