Question 1

(i)

Let $L_1 = \{a^i \vert i \geq 0 \}$ and $L_2 = \{b^i \vert i \geq 0 \}$. Then $L_1 \bowtie L_2 = \{a^ib^i \vert i \geq 0 \}$. L₁ and L₂are regular but $L_1 \bowtie L_2$ is not regular. Hence the statement is false.

(ii)

Let $L_1 = \{a^ib^i \vert i \geq 0 \}$ and $L_2 = \{c^i \vert i \geq 0 \}$. Then $L_1 \bowtie L_2 = \{a^ib^ic^{2i} \vert i \geq 0 \}$. L₁ and L₂are context-free but $L_1 \bowtie L_2$ is not context-free. Hence the statement is false.