Question

Consider the following statements regarding grammars.
I. If a grammar G has $ϵ$ productions. then $ϵ$ is a member of L(G).
II. If a grammar G has no $ϵ$ productions, then $ϵ$ is not a member of L(G).
III. If $ϵ$ is not a member of L(G). then G has on $ϵ$ productions.

Which of the above statements are correct

• A. I II and III
• B. II and III only
• C. II only
• D. I and II

Answer 1

The correct option is C II only

I is false For example, take $s\to aA:A\to ϵ$.
Clearly, G has a $ϵ$ production. however $ϵ$ is still not a member of L(G) as G only generates the string "a".
III is the contrapostive of I. hence III is also false .
But II correct. The presence of an $ϵ$ production is obviously a necessary condition for $ϵ$ to be a member of L(G).