Context Free Language Practice Questions

If L1 is context free language and L2 is a regular language which of the following is/are false?

If $L_1$ and $L_2$ are context free languages and R a regular set, one of the languages below is not necessarily a context free language. Which one?

Define a context free languages $L \in \{0, 1\}^*, \text{init} (L) = \{u \mid uv \in L$ for some v in $\{0, 1\}^*\}$ ( in other words, $\text{init}(L)$ is the set of prefixes of L) Let L = \{w \mid w \text{ is nonempty and has an equal number of 0$'s and$1's}\} Then $\text{init}(L)$ is:

Which of the following definitions below generate the same language as $L$ , where $L=\{x^ny^n \text{ such that } n\geq 1 \}$ ? I. $E \rightarrow xEy\mid xy$ II. $x y \mid (x^+xyy^+)$ III. $x^+y^+$

Choose the correct alternatives (more than one may be correct) and write the corresponding letters only: Context-free languages are:

Context-free languages and regular languages are both closed under the operation (s) of :