$(2$ pts$)$ Design a context$-$free grammar $($CFG$)$ for the language

over the alphabet

such that

consists of all strings where the number of

$\text{'}$s is twice the number of

$\text{'}$s$.$

$2.(8$ pts$)$ Consider the context$-$free grammar defined by:

I. Identify the variables, terminals, and start symbol of this grammar.

II$.$ Describe the language generated by this grammar.

III. Provide a derivation for the string "ababb".

IV$.$ Is the string "ababab" in the language? Justify your answer.

$3.(5$ pts$)$ Define a pushdown automaton $($PDA$)$ for the language

:

I. List the states, input alphabet, stack alphabet, start state, and transition functions of the PDA.

II$.$ Explain how the PDA processes the string $"0011".$

III. Illustrate the PDA's operation for the string $"000111"$ step$-$by$-$step.

$4.(5$ pts$)$ Prove or disprove the following statement: The language

is context$-$free. Provide a detailed explanation or proof.

$5.(5$pts$)$ Design a context$-$free grammar $($CFG$)$ that generates all strings over

that contain at least one $\text{'}$a$\text{'},$ one $\text{'}$b$\text{'},$ and one $\text{'}$c$\text{'}.$

I. Define the rules of the CFG$.$

II$.$ Provide a derivation for the string "abc".

III. Explain why the CFG does not generate the string $"$ab$"$ or $"$bc$".$