For any undecidable language \( L_1 \) and decidable language \( L_2 \), is their union \( L_1 \cup L_2 \) undecidable? Prove it.

Question

18-06-2025
Computers and Technology

High School

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For any undecidable language \( L_1 \) and decidable language \( L_2 \), is their union \( L_1 \cup L_2 \) undecidable? Prove it.

Answer :

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chriskel chriskel · Answer 1 · 2023-09-26T10:15:48-04:00

Final answer:

The union of an undecidable language L1 and a decidable language L2 can be undecidable because no algorithm can determine membership for every string in L1 and, by extension, in the union L1 ∪ L2.

Explanation:

The student asked whether the union of an undecidable language L1 and a decidable language L2 is undecidable. We approach the problem by looking at the properties of decidable and undecidable languages. If L1 is undecidable, there is no algorithm that can determine membership for every string in L1. However, for L2, such an algorithm exists. When considering the union L1 ∪ L2, for any string, the algorithm for L2 can determine whether it belongs to L2. If it does, then the string also belongs to the union. If not, the undecidability of L1 implies that we cannot decisively determine membership for the union through an algorithm, because we are not guaranteed to derive the answer from L1. Therefore, the union of an undecidable language and decidable language can indeed be undecidable.

Conclusion: The union L1 ∪ L2 is undecidable because the presence of an undecidable language L1 does not allow us to construct an algorithm that can decide membership for every element in the union.