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Is this given answer correct?

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yes, given answer is correct,

R' ∩ C' = ( R U C )' = C' ===> is it CFL closed under complementation? No ===> Is it CSL closed under Complementation? yes

therefore it must be CSL but it may or may not CFL

When CFL is DCFL then it is closed under complementation
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R is regular so complement(R) is also regular as regular language are closed under complementation .

C is cfl so complement (C) is necessarily not cfl as cfl are not closed under complementation bt its is necessarily csl becoz cfl is subset of csl.

so intersection of csl and regular is definitely csl .(becoz regular are also csl and csl ∩ csl = csl as csl are closed under intersection)....

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L1=regular so L1(complement) is also regular as well as CFL.

L2=CFL so its complement will be CSL .now $L1\bigcap L2$ will be CSL not CFL
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it may be CFL

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