Question

Consider the following statements:
I. If L is a regular language then the set of strings in L of odd length is also a regular language.
II. If L is a regular language then the set of strings in L of even length is also a regular language.
Which of the above statements is/are true?

• A. None of these
• B. Both I and II
• C. Only I
• D. Only II

Answer 1

The correct option is B Both I and II

Let L be any language and let Even(L) and Odd (L) be
Even (L) = the set of strings in L of even string.
Odd (L) = the set of strings in L of odd string.
Then note that Even(L) = L$\cap$Even$\sum$*
Also note that Odd(L) = L$\cap$ Odd $\sum$*

So, if L is regular then Even(L) and Odd(L) will also be
both regular and regular languages are closed under intersection.