Equivalence Class of Empty Set

Question

Im trying to solve this problem, but i'm really stuck.

Let $(\Omega, \mathcal{B}, \mathbb{P})$ a probability space with the equivalence relation $ X \sim Y \iff \mathbb{P}(X \bigtriangleup Y) = 0$ over $\mathcal{B}$ where $ \bigtriangleup$ is the symmetric difference.

Show that $[\emptyset] \cup [\Omega] = \{X \in \mathcal{B} : \mathbb{P}(X) = 0 ~\text{or} ~\mathbb{P}(X) = 1\}$ where $[\cdot]$ is the equivalence class.

Answer 1

Hint:

$\ \emptyset\bigtriangleup Y=Y\ $ and $\ \Omega \bigtriangleup Y=\Omega\setminus Y\ $ for all $\ Y\in\cal{B}\ $.

So if $\ P(\emptyset\bigtriangleup Y)=0\ $ for all $\ Y\in[\emptyset]\ $, what does that tell you about the value of $\ P(Y)\ $ for $\ Y\in[\emptyset]\ $?

And if $\ P(\Omega\bigtriangleup Y)=0\ $ for all $\ Y\in[\Omega]\ $, what does that tell you about the value of $\ P(Y)\ $ for $\ Y\in[\Omega]\ $?