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


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 '17 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 '17 at 11:16

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