Identify the class of L

Question

Consider

$L_1 = \left\{a^nb^nc^md^m \mid m,n \ge 1\right\}$

$L_2 = \left\{a^nb^n \mid n \ge1\right\}$

$L_3 = \left\{(a+b)^*\right\}$


$L_1$ - $L_3$ is

(A) Regular (B) CFL but not regular (C) CSL but not CFL (D) None of these

Answer 1

CFL

 $L_1 - L_3 = L_1$, hence CFL
 Proof,
 $L_1 - L_3 = \{abcd,aabbcd,aaabbbccdd,\dots\} - \{\epsilon ,a,b,ab,aab,\dots\}$
 $= \{abcd,aabbcd,aaabbbccdd,\dots\}$
 $= L_1$

Comments

but ∅ means regular then L2-L3 = regular too

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yes, it is regular.

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{a^n.bⁿ - (a+b)^* } = Reguler because L2 is subset of L3 

and if  i follow clousre table then answer is CFL.

Answer 2

L1 - L3 =L1 $\bigcap$ L3' = L1 $\bigcap$ $\phi$ = $\phi$= Regular

So, option A Regular should be ans.

Note :- L3=$\sum$* So L3'=$\phi$

Answer 3

L1 is DCFL which is also CFL

L2 is DCFL which is also CFL

and L3 is Reguler then L1 - L3 = L1

Therefore option B is Correct..