Which of the following statement(s) is/are TRUE while computing $First$ and $Follow$ during top down parsing by a compiler?

To determine the truth value of each statement:

1. Option A is TRUE: By definition, if $A \to \epsilon$ is a production in the grammar, then $\epsilon$ is added to $\text{First}(A)$.

2. Option B is FALSE: The input right end marker (typically represented as \$) is placed in the$$\text{Follow}$$set of the start symbol, not its$$\text{First}$$ set.

3. Option C is FALSE: The empty string $\epsilon$ is never an element of any $\text{Follow}$ set.

4. Option D is TRUE: By standard definition of $\text{Follow}$, \$(the input right end marker) is placed in$$\text{Follow}$(S)$, where$S$ is the start symbol of the grammar.

Thus, statements A and D are correct.