# Infinite Union of Recursive language

## 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

• sir, that means $\text{none of regular,cfl,csl,recursive and recursive enumerable are closed under Infinite union}$ – virat 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