Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Which of the following options is INCORRECT?a...
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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 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.
Most Upvoted Answer
Which of the following options is INCORRECT?a)If L is a recursive lang...
Understanding Recursive Languages
When evaluating the options, it’s crucial to understand the definitions and properties of recursive and recursively enumerable languages, among other concepts in set theory.
Option A: Recursive Languages
- If L is a recursive language, it is always recursive. This statement is true by definition, as recursive languages are a subset of recursively enumerable languages.
Option B: Closure Properties
- The second option states that if L and its complement L are both recursively enumerable, then both are recursive. This is also true, as it follows from the closure properties of recursively enumerable languages.
Option C: Union of Languages
- The union of two recursively enumerable languages is recursively enumerable. This property holds, making this statement true as well.
Option D: Cartesian Product of Sets
- The statement that the Cartesian product of two countable sets is uncountable is incorrect. In fact:
- The Cartesian product of two countable sets is countable.
- Countable sets can be finite or infinite, and when you take a product of two countable sets, the resulting set is still countable.
Conclusion
Thus, the correct answer is option D, as it incorrectly asserts that the Cartesian product of two countable sets is uncountable, which contradicts established principles in set theory.
When evaluating the options, it’s crucial to understand the definitions and properties of recursive and recursively enumerable languages, among other concepts in set theory.
Option A: Recursive Languages
- If L is a recursive language, it is always recursive. This statement is true by definition, as recursive languages are a subset of recursively enumerable languages.
Option B: Closure Properties
- The second option states that if L and its complement L are both recursively enumerable, then both are recursive. This is also true, as it follows from the closure properties of recursively enumerable languages.
Option C: Union of Languages
- The union of two recursively enumerable languages is recursively enumerable. This property holds, making this statement true as well.
Option D: Cartesian Product of Sets
- The statement that the Cartesian product of two countable sets is uncountable is incorrect. In fact:
- The Cartesian product of two countable sets is countable.
- Countable sets can be finite or infinite, and when you take a product of two countable sets, the resulting set is still countable.
Conclusion
Thus, the correct answer is option D, as it incorrectly asserts that the Cartesian product of two countable sets is uncountable, which contradicts established principles in set theory.
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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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