Consider the following languages: $$L_1 = \{ww \mid w \in \{a,b\}^*\}$$ $$L_2 = \{a^n b^n c^m \mid m, n \ge 0\}$$ $$L_3 = \{a^m b^n c^n \mid m, n \ge 0\}$$ Which of the following statements is/are FALSE?

Let us analyze each language and statement:

1. Language properties:

   • $L_1 = \{ww \mid w \in \{a,b\}^*\}$ is a well-known non-context-free language (provable by the Pumping Lemma for CFLs). However, its complement $\overline{L_1}$ is context-free because any string not in $L_1$ is either of odd length, or of even length $2k$ where there exists some position $i$ ($1 \le i \le k$) such that the $i$-th symbol differs from the $(k+i)$-th symbol.

   • $L_2 = \{a^n b^n c^m \mid m, n \ge 0\}$ is a Deterministic Context-Free Language (DCFL) because a DPDA can push all $a$'s, match and pop with $b$'s, and then consume $c$'s.

   • $L_3 = \{a^m b^n c^n \mid m, n \ge 0\}$ is also a DCFL.

   • $L_2 \cap L_3 = \{a^n b^n c^n \mid n \ge 0\}$, which is well known to be non-context-free.

2. Evaluating the options:

   • Statement (A): "$L_1$ is not context-free but $L_2$ and $L_3$ are deterministic context-free." — This statement is TRUE.

   • Statement (B): "Neither $L_1$ nor $L_2$ is context-free." — This statement is FALSE because $L_2$ is context-free.

   • Statement (C): "$L_2$, $L_3$ and $L_2 \cap L_3$ all are context-free." — This statement is FALSE because $L_2 \cap L_3$ is not context-free.

   • Statement (D): "Neither $L_1$ nor its complement is context-free." — This statement is FALSE because the complement of $L_1$ is context-free.

Since the question asks which statements are FALSE, the correct choices are B, C, and D.