Is the BPP class closed for union and intersection?

Question

Just like the title says. I want to prove that given two languages $L_1,L_2 \in BPP$, $L_1 \cup L_2 \in BPP$ and $L_1 \cap L_2 \in BPP$

Answer 1

Let $T_1$ (resp. $T_2$) a polynomial-time probabilistic Turing machine with error probability of at most $1/3$ for $L_1$ (resp. $L_2$).

Let $T'_1$ (resp. $T'_2$) be a Turing machine for $L_1$ (resp. $L_2$) obtained by running $T_1$ (resp. $T_2$) $9$ times and returning the majority result. The error probability of $T'_1$ (resp. $T'_2$) is at most $p_F = \sum_{i=5}^9 \binom{9}{i} (1/3)^i (2/3)^{9-i} < \frac{1}{6}$.

A Turing machine for $L_1 \cup L_2$ (resp. $L_1 \cap L_2$) simulates $T'_1$ and $T'_2$ and accepts if and only if at least one of (resp. both of) $T'_1$ and $T'_2$ accept. By the union bound, the probability of error of this Turing machine is at most $2 \cdot p_F < 2 \cdot \frac{1}{6} = \frac{1}{3}$.