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Consider the ...

Question

Consider the following types of languages $L_{1} : R e g u l a r, L_{2} : C o n t e x t - f r e e, L_{3} : R e c u r s i v e, L_{4} : R e c u r s i v e l y e n u m e r a b l e .$
Which of the following is/are TRUE?
I. $_{3} \cup L_{4}$ is recursively enumerable.
II. $_{2} \cup L_{3}$ is recursive.
III. $_{1} \cap L_{2}$ is context-free.
IV. $L_{1} \cup_{2}$ is context-free.

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Solution

$L_{1} \to$ Regular, $L_{2} \to$ CFL
$L_{3} \to$ REC
$L_{4} \to$ RE
I. ${¯ L}_{3} \cup L_{4}$ is RE
${¯ L}_{3} \cup L_{4} = ¯ ¯¯¯¯¯¯¯¯¯¯ ¯ R E C \cup R E = R E C \cup R E$
=RE $\cup R E = R E$
So I is true
II. ${¯ L}_{2} \cup L_{3}$ is recursive
${¯ L}_{2} \cup L_{3} = ¯ ¯¯¯¯¯¯¯¯¯¯ ¯ C F L \cup R E C = ¯ ¯¯¯¯¯¯¯¯¯ ¯ C S L \cup R E C$
=CSL $\cup R E C$
=REC $\cup R E C = R E C$
So II is True.
III. $L_{1}^{} \cap L_{2}$ is CFL

$L_{1}^{} \cap L_{2}$ = $(R E G)^{*} \cap C F L$
$= R E G \cap C F L$ =CFL
So III is True
IV. $L_{1} \cup {¯ L}_{2}$ is CFL
$L_{1} \cup {¯ L}_{2} = R E G \cup ¯ ¯¯¯¯¯¯¯¯¯¯ ¯ C F L = R E G \cup ¯ ¯¯¯¯¯¯¯¯¯ ¯ C S L$
$= R E G \cup C S L$ =CSL
Since, A CSL need not be a CFL,So IV is False
So only I,II and III are true.

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