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Consider the following statements: 1. The set of all irrational numbers between 12 and 15 is an infinite set. II. The set of all odd integers less than 1000 is a finite set. Which of the statements given above is/are correct?
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Consider the following statements: 1. The set of all irrational number...
Analysis of Statements
To determine the correctness of the given statements, we need to analyze each one carefully.
Statement I: The set of all irrational numbers between 12 and 15 is an infinite set.
- Definition of Irrational Numbers: Irrational numbers are those that cannot be expressed as a fraction of two integers. Examples include numbers like √2, π, and e.
- Interval of Interest: The interval between 12 and 15 is continuous. Within any interval of real numbers, there are both rational and irrational numbers.
- Density of Irrational Numbers: Between any two real numbers, there are infinitely many irrational numbers. Thus, the set of all irrational numbers between 12 and 15 is indeed infinite.
Statement II: The set of all odd integers less than 1000 is a finite set.
- Definition of Odd Integers: Odd integers are those integers that are not divisible by 2. Examples include -1, 1, 3, 5, etc.
- Range of Interest: The set of odd integers less than 1000 includes numbers starting from 1 up to 999.
- Count of Odd Integers: The odd integers in this range can be expressed as 1, 3, 5,..., 999. This is an arithmetic sequence where the first term is 1 and the last term is 999, with a common difference of 2.
- Finite Count: The total count of odd integers less than 1000 is finite, specifically there are 500 odd integers (from 1 to 999).
Conclusion
- Correctness of Statements:
- Statement I is correct.
- Statement II is correct.
Thus, both statements are true.
To determine the correctness of the given statements, we need to analyze each one carefully.
Statement I: The set of all irrational numbers between 12 and 15 is an infinite set.
- Definition of Irrational Numbers: Irrational numbers are those that cannot be expressed as a fraction of two integers. Examples include numbers like √2, π, and e.
- Interval of Interest: The interval between 12 and 15 is continuous. Within any interval of real numbers, there are both rational and irrational numbers.
- Density of Irrational Numbers: Between any two real numbers, there are infinitely many irrational numbers. Thus, the set of all irrational numbers between 12 and 15 is indeed infinite.
Statement II: The set of all odd integers less than 1000 is a finite set.
- Definition of Odd Integers: Odd integers are those integers that are not divisible by 2. Examples include -1, 1, 3, 5, etc.
- Range of Interest: The set of odd integers less than 1000 includes numbers starting from 1 up to 999.
- Count of Odd Integers: The odd integers in this range can be expressed as 1, 3, 5,..., 999. This is an arithmetic sequence where the first term is 1 and the last term is 999, with a common difference of 2.
- Finite Count: The total count of odd integers less than 1000 is finite, specifically there are 500 odd integers (from 1 to 999).
Conclusion
- Correctness of Statements:
- Statement I is correct.
- Statement II is correct.
Thus, both statements are true.
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Consider the following statements: 1. The set of all irrational numbers between 12 and 15 is an infinite set. II. The set of all odd integers less than 1000 is a finite set. Which of the statements given above is/are correct?
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Consider the following statements: 1. The set of all irrational numbers between 12 and 15 is an infinite set. II. The set of all odd integers less than 1000 is a finite set. Which of the statements given above is/are correct? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Consider the following statements: 1. The set of all irrational numbers between 12 and 15 is an infinite set. II. The set of all odd integers less than 1000 is a finite set. Which of the statements given above is/are correct? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statements: 1. The set of all irrational numbers between 12 and 15 is an infinite set. II. The set of all odd integers less than 1000 is a finite set. Which of the statements given above is/are correct?.
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