Consider the following grammar 𝐺, with 𝑆 as the start symbol. The grammar 𝐺…

Final filled productions:

• S → R f

• T → ε

• R → c T R

Reasoning:

• S: The grammar already provides S → d a T which covers starting symbol d. To obtain the remaining FIRST symbols c and f, using S → R f works because FIRST(R) = {c, ε}. If R derives a string starting with c then S can start with c; if R → ε then S starts with f. Together with the existing d production this yields FIRST(S) = {c, d, f}.

• T: The productions T → a S and T → b T already give a and b in FIRST(T). Adding T → ε is necessary so that FIRST(T) = {a, b, ε} matches the given set.

• R: With R → c T R and R → ε available, FIRST(R) = {c, ε} as required: the c production provides c, and the ε alternative provides ε.

Why other forms fail:

• Placing f at the start of the S production (for example S → f R) prevents S from starting with c, so FIRST(S) would miss c and not match the given FIRST(S).

• Using a production like T → c T would introduce c into FIRST(T), which contradicts the given FIRST(T) = {a, b, ε}.

• Defining R without an ε alternative would remove ε from FIRST(R), contradicting the given FIRST(R) = {c, ε}.