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Which of the following options is INCORRECT?
  • a)
    If L is a recursive language, L' is also recursive.
  • b)
    If L and L' both are recursively enumerable, L and L' are recursive.
  • c)
    Union of two recursively enumerable languages is recursively enumerable.
  • d)
    The cartesian product of two countable sets is uncountable.
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Which of the following options is INCORRECT?a)If L is a recursive lang...
Option 4 is incorrect. It can be corrected as:
The cartesian product of two countable sets is countable.
Proof: There are three cases to consider:
Case 1: If both A and B are finite with |A|=m and |B|=n, then it is easy to show that |A×B|=mn and hence A×B is finite and so it is countable.
Case 2: If A is finite with |A|=n and B is countably infinite, there exist bijective functions f: A→{1,2,...,n} and g: B→N. We can define h: A×B→N for all (a,b)∈A×B.
Then clearly h is injective. So A×B is countable.
Case 3: If A and B are both countable, there exist bijective functions f: A→N and g: B→N and defining h: A×B→N gives us an injective function, and so A×B is countable. 
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Which of the following options is INCORRECT?a)If L is a recursive language, L' is also recursive.b)If L and L' both are recursively enumerable, L and L' are recursive.c)Union of two recursively enumerable languages is recursively enumerable.d)The cartesian product of two countable sets is uncountable.Correct answer is option 'D'. Can you explain this answer?
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Which of the following options is INCORRECT?a)If L is a recursive language, L' is also recursive.b)If L and L' both are recursively enumerable, L and L' are recursive.c)Union of two recursively enumerable languages is recursively enumerable.d)The cartesian product of two countable sets is uncountable.Correct answer is option 'D'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Which of the following options is INCORRECT?a)If L is a recursive language, L' is also recursive.b)If L and L' both are recursively enumerable, L and L' are recursive.c)Union of two recursively enumerable languages is recursively enumerable.d)The cartesian product of two countable sets is uncountable.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following options is INCORRECT?a)If L is a recursive language, L' is also recursive.b)If L and L' both are recursively enumerable, L and L' are recursive.c)Union of two recursively enumerable languages is recursively enumerable.d)The cartesian product of two countable sets is uncountable.Correct answer is option 'D'. Can you explain this answer?.
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