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


1 Answer 1


Every finite language is recursive.

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

Some language isn't recursive.

  • $\begingroup$ sir, that means $\text{none of regular,cfl,csl,recursive and recursive enumerable are closed under Infinite union}$ $\endgroup$ Oct 15, 2017 at 11:08
  • $\begingroup$ 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. $\endgroup$ Oct 15, 2017 at 11:16

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