Consider the 5-state DFA $M$ accepting the language $L(M) \subset (0 + 1)^*$ shown below. For any string $w \in (0 + 1)^*$ let $n_0(w)$ be the number of 0's in $w$ and $n_1(w)$ be the number of 1's in $w$. Which of the following statements is/are FALSE?

Let us analyze the transitions of the given DFA $M$:

• State $1$ is both the start state and the only accepting state: $F = \{1\}$.

• $\delta(1, 0) = 2$, $\delta(1, 1) = 4$

• $\delta(2, 0) = 3$, $\delta(2, 1) = 1$

• $\delta(3, 0) = 1$, $\delta(3, 1) = 2$

• $\delta(4, 0) = 1$, $\delta(4, 1) = 5$

• $\delta(5, 0) = 4$, $\delta(5, 1) = 1$

### 1. State Distinguishability / Equivalence

• Pair $(2, 5)$:

  • On input $0$: $\delta(2, 0) = 3$ and $\delta(5, 0) = 4$.

  • On input $1$: $\delta(2, 1) = 1 \in F$ and $\delta(5, 1) = 1 \in F$.

• Pair $(3, 4)$:

  • On input $0$: $\delta(3, 0) = 1 \in F$ and $\delta(4, 0) = 1 \in F$.

  • On input $1$: $\delta(3, 1) = 2$ and $\delta(4, 1) = 5$.

Since both transitions from $\{2, 5\}$ lead to $\{3, 4\}$ on $0$ and $\{1\}$ on $1$, and both transitions from $\{3, 4\}$ lead to $\{1\}$ on $0$ and $\{2, 5\}$ on $1$, states $2$ and $5$ are indistinguishable ($2 \equiv 5$), and states $3$ and $4$ are indistinguishable ($3 \equiv 4$).

• Statement (B) says states $3$ and $4$ are distinguishable, which is FALSE (they are equivalent).

• Statement (C) says states $2$ and $5$ are distinguishable, which is FALSE (they are equivalent).

### 2. Pair $(2, 4)$

• On input $1$: $\delta(2, 1) = 1 \in F$, but $\delta(4, 1) = 5 \notin F$.

• Thus, string 1 distinguishes states $2$ and $4$. Statement (A) is TRUE.

### 3. Language of the DFA The states track the value of $(n_0(w) - n_1(w)) \pmod 3$:

• State $1$: $(n_0(w) - n_1(w)) \equiv 0 \pmod 3$

• States $\{2, 5\}$: $(n_0(w) - n_1(w)) \equiv 1 \pmod 3$

• States $\{3, 4\}$: $(n_0(w) - n_1(w)) \equiv 2 \pmod 3$

If $n_0(w) = n_1(w)$, then $n_0(w) - n_1(w) = 0 \equiv 0 \pmod 3$, which leads to state $1$ (the accepting state). Therefore, any such string is accepted. Statement (D) is TRUE.

Since the question asks for the FALSE statements, the correct options are B and C.