Let $X$ be a set of cardinality $n$ and let $R$ be the set of all relations over $X$. Consider the probability space $(R,P)$ with uniform distribution, that is, $P[\{R_1\}]] = P[\{R_2\}]$ for all $ R_1,R_2 \in R$. Define the relation ”equiprobable” on the set of all events of $(R, P)$ as follows: Events $A$ and $B$ are equiprobable iff $P[A] = P[B]$.
a. Prove that ”equiprobable” is an equivalence relation.
b. How many equivalence classes does this relation define?
Here is my attempt at a, can anyone help me check?
Reflexive: $R_1 ,\in R$, then $P[\{R_1\}] = P[\{R_1\}]$, which is equiprobable $\Rightarrow$ $R_1\,R\, R_1$ $\Rightarrow$ reflexive
Transitive: $R_1,R_2,R_3 \in R$, suppose $P[\{R_1\}] = P[\{R_2\}\,] \& \,P[\{R_2\}] = P[\{R_3\}] \,\Rightarrow\, P[\{R_1\}] = P[\{R_3\}] \,\Rightarrow\, R_1 \,R \,R_3 \,\Rightarrow $transitive
Symmetric: $R_1, R_2 \in R$, $ P[\{R_1\}] = P[\{R_2\}]$ $\Rightarrow$ $P[\{R_2\}] = P[\{R_1\}]$ $\Rightarrow$ $R_2\, R\, R_1$ $\Rightarrow$ symmetric
I don't know how to solve b, and my teacher gave me a hint but I'm still a lot confused about equivalence classes
If $R_1$ and $R_2$ are equiprobable, is there any relation between $|R_1|$ and $|R_2|$?
This is just my guess: $|R_1| = |R_2|$ because they are equivalence relations, so the size of them are equal? Please help me understand this