I'm trying to show this property of the symmetric difference between two sets defined for two sets in a universe $A$ and $B$ by
$$
A\Delta B=(A\cap B^{c})\cup(B\cap A^{c})
$$
I need to show that
$$
\mathbb{P}(A\Delta C)\leq \mathbb{P}(A\Delta B)+\mathbb{P}(B\Delta C)
$$
for sets $A, B,$ and $C$ in the universe.
I showed in the first part of the problem that
$$
\mathbb{P}(A\Delta B)=\mathbb{P}(A)+\mathbb{P}(B)-2\mathbb{P}(A\cap B)
$$
My idea was to note that
$$
\mathbb{P}(A\Delta C)\leq\mathbb{P}(A\cap C^{c})+\mathbb{P}(C\cap A^{c})
$$
by probability laws and then leverage the fact that for any set I can write it as a union with another set. That is
$$
\mathbb{P}(A)=\mathbb{P}(A\cap B)+\mathbb{P}(B^{c}\cap A)
$$
and likewise for $C$ to substitute in for $P(A)$ and $P(B)$ terms. However, I end up running in circles. My TA did say I was on the right track, though. Any suggestions would be helpful. Thanks.
Best Answer
If $\chi_U$ denotes the characteristic function of the set $U$, then we have $\chi_{A\Delta B} = |\chi_A - \chi_B|$. Hence we have \begin{align*} \mathbb{P}(A\Delta C) &= \int |\chi_A - \chi_C| d\mu \\ &= \int |(\chi_A - \chi_B) + (\chi_B - \chi_C)| d\mu \\ &\leq \int (|\chi_A - \chi_B| + |\chi_B - \chi_C|) d\mu \\ &= \int (|\chi_A - \chi_B|d\mu + \int |\chi_B - \chi_C|) d\mu\\ &= \mathbb{P}(A\Delta B) + \mathbb{P}(B\Delta C) \end{align*}