Intersection and concatenation in CFG

Question

Given that L1 is regular and L2 context free.

i) L3 = L1 ∩ L2

ii) L4= L1.L2

Selct the most appropriate statement:

a. L3 , L4 are regular

b. L3 is regular L4 is CFG not regular

c. L3 is CFG, not regular L4 is regular

d. L3,L4 are CFG not regular

Comments

is it D?

Answer 1

Option D) is almost correct.

 1) Intersection of CFL and regular is CFL but it may be regular too. For eg if L1 is $\epsilon$ which is regular then intersection of L1 with any langauage will be $\epsilon$ which is regular. So it is defintely CFL and may or may not be regular

2) Similarly concatenation is also closed for CFL. but in certain cases it may be regular. If L1 = $\phi$ then concatenation with always be phi which is regular.

So both are definetely CFL and may or may not be regular.