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Let N be the set of natural numbers. Consider the following sets, P: Set of Rational numbers (positive and negative) Q: Set of functions from {0, 1} to N R: Set of functions from N to {0, 1} S: Set of finite subsets of N Which of the above sets are countable?
- a)Q and S only
- b)P and S only
- c)P and R only
- d)P, Q and S only
Correct answer is option 'D'. Can you explain this answer?
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Let N be the set of natural numbers. Consider the following sets, P: S...
Set of Rational numbers (+ve or -ve) are countable.. Set of functions from {0, 1} to N are countable because it has one to one correspondence to N. Set of functions from N to {0, 1} is uncountable, because it has one to one correspondence to set of real numbers between (0 and 1). Set of finite subsets of N is countable. Sets P, Q and S are countable, therefore option (D) is Correct.
Most Upvoted Answer
Let N be the set of natural numbers. Consider the following sets, P: S...
The correct answer is option 'D', which means that the sets P, Q, and S are countable. Let's analyze each set to understand why they are countable.
1. Set P: Set of Rational numbers (positive and negative)
- Rational numbers can be represented as fractions, where the numerator and denominator are integers.
- We can list all the rational numbers by enumerating the fractions in a systematic way, such as by listing them in increasing order of their absolute values.
- Since the set of integers is countable, and every rational number can be expressed as a fraction of two integers, the set of rational numbers is countable.
2. Set Q: Set of functions from {0, 1} to N
- A function from {0, 1} to N can be represented as a pair of natural numbers.
- We can list all the possible pairs of natural numbers in a systematic way, such as by listing them in increasing order of their sum.
- Since the set of natural numbers is countable, the set of functions from {0, 1} to N is countable.
3. Set R: Set of functions from N to {0, 1}
- A function from N to {0, 1} can be thought of as a sequence of 0s and 1s.
- We can list all the possible sequences of 0s and 1s in a systematic way, such as by listing them in lexicographic order (dictionary order).
- Since the set of sequences of 0s and 1s is equivalent to the set of binary numbers, which is countable, the set of functions from N to {0, 1} is countable.
4. Set S: Set of finite subsets of N
- A finite subset of N can be represented as a sequence of natural numbers.
- We can list all the possible finite sequences of natural numbers in a systematic way, such as by listing them in lexicographic order.
- Since the set of finite sequences of natural numbers is countable, the set of finite subsets of N is countable.
In conclusion, the sets P, Q, and S are countable because they can be enumerated in a systematic way. Therefore, the correct answer is option 'D' - P, Q, and S only.
1. Set P: Set of Rational numbers (positive and negative)
- Rational numbers can be represented as fractions, where the numerator and denominator are integers.
- We can list all the rational numbers by enumerating the fractions in a systematic way, such as by listing them in increasing order of their absolute values.
- Since the set of integers is countable, and every rational number can be expressed as a fraction of two integers, the set of rational numbers is countable.
2. Set Q: Set of functions from {0, 1} to N
- A function from {0, 1} to N can be represented as a pair of natural numbers.
- We can list all the possible pairs of natural numbers in a systematic way, such as by listing them in increasing order of their sum.
- Since the set of natural numbers is countable, the set of functions from {0, 1} to N is countable.
3. Set R: Set of functions from N to {0, 1}
- A function from N to {0, 1} can be thought of as a sequence of 0s and 1s.
- We can list all the possible sequences of 0s and 1s in a systematic way, such as by listing them in lexicographic order (dictionary order).
- Since the set of sequences of 0s and 1s is equivalent to the set of binary numbers, which is countable, the set of functions from N to {0, 1} is countable.
4. Set S: Set of finite subsets of N
- A finite subset of N can be represented as a sequence of natural numbers.
- We can list all the possible finite sequences of natural numbers in a systematic way, such as by listing them in lexicographic order.
- Since the set of finite sequences of natural numbers is countable, the set of finite subsets of N is countable.
In conclusion, the sets P, Q, and S are countable because they can be enumerated in a systematic way. Therefore, the correct answer is option 'D' - P, Q, and S only.
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Let N be the set of natural numbers. Consider the following sets, P: Set of Rational numbers (positive and negative) Q: Set of functions from {0, 1} to N R: Set of functions from N to {0, 1} S: Set of finite subsets of N Which of the above sets are countable?a)Q and S onlyb)P and S onlyc)P and R onlyd)P, Q and S onlyCorrect answer is option 'D'. Can you explain this answer?
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Let N be the set of natural numbers. Consider the following sets, P: Set of Rational numbers (positive and negative) Q: Set of functions from {0, 1} to N R: Set of functions from N to {0, 1} S: Set of finite subsets of N Which of the above sets are countable?a)Q and S onlyb)P and S onlyc)P and R onlyd)P, Q and S onlyCorrect 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 Let N be the set of natural numbers. Consider the following sets, P: Set of Rational numbers (positive and negative) Q: Set of functions from {0, 1} to N R: Set of functions from N to {0, 1} S: Set of finite subsets of N Which of the above sets are countable?a)Q and S onlyb)P and S onlyc)P and R onlyd)P, Q and S onlyCorrect 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 Let N be the set of natural numbers. Consider the following sets, P: Set of Rational numbers (positive and negative) Q: Set of functions from {0, 1} to N R: Set of functions from N to {0, 1} S: Set of finite subsets of N Which of the above sets are countable?a)Q and S onlyb)P and S onlyc)P and R onlyd)P, Q and S onlyCorrect answer is option 'D'. Can you explain this answer?.
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