If $L_1 = \{a^n \mid n \ge 0\}$ and $L_2 = \{b^n \mid n \ge 0\}$, consider (I) $L_1 \cdot L_2$ is a regular language (II) $L_1 \cdot L_2 = \{a^n b^n \mid n \ge 0\}$ Which one of the following is CORRECT?

Given:

• $L_1 = \{a^n \mid n \ge 0\} = L(a^*)$ is a regular language.

• $L_2 = \{b^n \mid n \ge 0\} = L(b^*)$ is a regular language.

Evaluating the statements:

1. Statement (I): The concatenation of two regular languages is always regular (closure property). Here, $L_1 \cdot L_2 = L(a^*b^*)$, which is regular. Thus, statement (I) is TRUE.

2. Statement (II): By definition of concatenation of languages:

The powers of $a$ and $b$ are chosen independently. Hence, $L_1 \cdot L_2 \neq \{a^n b^n \mid n \ge 0\}$. Thus, statement (II) is FALSE.

Therefore, Only (I) is correct, which corresponds to option (A).