GATE CS 2013 Q.8 (CS Q.8): Regular Expressions, Finite Automata and Regular Languages

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

$\{\epsilon\}$

$\Phi$

$a^*$

$\{\epsilon, a\}$

Official answer and solution

Solution

Apply the definitions of concatenation and Kleene star for the empty language and singleton language.

Given $L_1 = \Phi = \emptyset$ and $L_2 = \{ a \}$ :

Concatenation with the empty set:
Concatenating any language with the empty language yields the empty language:
$L_1 L_2^* = \emptyset \cdot \{ a \}^* = \emptyset$
Kleene star of the empty set:
By definition of the Kleene star, $L^* = \bigcup_{k=0}^{\infty} L^k$ , and $L^0 = \{\epsilon\}$ for any language $L$ , including the empty language:
$L_1^* = \emptyset^* = \{\epsilon\}$
Union:
$L_1 L_2^* \cup L_1^* = \emptyset \cup \{\epsilon\} = \{\epsilon\}$

Thus, the resulting language is $\{\epsilon\}$ .

Final answer

Option (A)

Common trap

Thinking that $\emptyset^*$ is $\emptyset$ , whereas $\emptyset^* = \{\epsilon\}$ because the zero-length string is always included.

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