Which of the following statements is/are FALSE? 1. For every non-deterministic Turing machine, there exists an equivalent deterministic Turing machine. 2. Turing recognizable languages are closed under union and complementation. 3. Turing decidable languages are closed under intersection and complementation. 4. Turing recognizable languages are closed under union and intersection.

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Solution

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Official answer: C) 2 only

Turing Machines and Undecidability

Let us analyze each statement:

Statement 1 is TRUE: Non-deterministic Turing Machines (NDTMs) and Deterministic Turing Machines (DTMs) recognize the exact same class of languages (the recursively enumerable languages). Any NDTM can be simulated by an equivalent DTM.
Statement 2 is FALSE: Turing recognizable (recursively enumerable) languages are closed under union, intersection, concatenation, and Kleene star, but they are not closed under complementation.
Statement 3 is TRUE: Turing decidable (recursive) languages are closed under all standard boolean operations: union, intersection, and complementation.
Statement 4 is TRUE: Turing recognizable languages are closed under both union and intersection.

Since the question asks for the statement(s) that is/are FALSE, only statement 2 is false. Thus, (C) is the correct option.

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This Turing Machines and Undecidability previous year question is Q17 of the GATE CSE 2013 paper. Find more GATE CSE PYQs on Turing Machines and Undecidability in Theory of Computation, or browse all GATE CSE PYQs.

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Turing Machines and Undecidability in GATE CSE

Turing Machines and Undecidability has 30 GATE CSE PYQs across 20 papers from 2012 to 2026. 13 of them are 1-mark MCQs like this one, and GATE CSE 2013 has 2.

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