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Question

Is Infinite Union of Recursive language is Recursive?

I know it is already posted here, but the i am not getting answer also i want to know if my approach is correct.

My Approach/Doubt

$\text{let}\,\,L_1=abcd \,\,\,L_2=a^2b^2c^2d^2=aabbccdd,L_3=a^3b^3c^3d^3=aaabbbcccddd$

here $L_{1},L_{2},L_{3}\,....\text{are finite hence regular hence recursive}$

let $L_{Iu}=L_1 \,\cup\,L_2\,\cup\,L_3\,\cup........$

but $L_{Iu}=a^{n}b^{n}c^{n} \text{which is recursive }$ .

so can i say that Infinite Union of Recursive language is Recursive

Please help me out

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sourav_anand
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1 Answers1

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Every finite language is recursive.

Every language can be written as an infinite union of finite languages.

Some language isn't recursive.

Yuval Filmus
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  • sir, that means $\text{none of regular,cfl,csl,recursive and recursive enumerable are closed under Infinite union}$ – sourav_anand Oct 15 '17 at 11:08
  • Right. If a family of languages contains all singletons (i.e., ${w}$) and is closed under arbitrary union, then it either contains all languages or all languages except the empty one. – Yuval Filmus Oct 15 '17 at 11:16