Consider the languages $L_1 = \Phi$ and $L_2 = \{a\}$. Which one of the following represents $L_1 L_2^* \cup L_1^*$ ?

Given the languages:

• $L_1 = \Phi$

• $L_2 = \{a\}$

Now, compute each component:

1. The concatenation of the empty set $\Phi$ with any language is $\Phi$:

   $$L_1 L_2^* = \Phi \cdot (\{a\})^* = \Phi$$

2. The Kleene star of the empty language contains only the empty string:

   $$L_1^* = \Phi^* = \{\epsilon\}$$

Combining the two via union:

Hence, (A) is the correct option.