Mathematics Exam > Mathematics Questions > Which one of the following is not correct?a)T...
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Which one of the following is not correct?
- a)The set I has no limit point.
- b)The set hi has no limit point.
- c)No point of Q is a limit point.
- d)Every point of the set M is a limit point.
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Which one of the following is not correct?a)The set I has no limit poi...
The set l has no limit point, for a nbd 
of m ε l, contains no point of l other than m. Thus the derived set of l is the null set φ. The set N has no limit point for a nbd 
of m ε N, contains no points of N other than m. Thus, the derived set of N is null set φ. Every point of the set Q of rationals is a limit point, for between any two rationals there exist an infinity of rational.
Further every point of R is a limit point, for every nbd of any of its points contains an infinite member of R.
of m ε l, contains no point of l other than m. Thus the derived set of l is the null set φ. The set N has no limit point for a nbd of m ε N, contains no points of N other than m. Thus, the derived set of N is null set φ. Every point of the set Q of rationals is a limit point, for between any two rationals there exist an infinity of rational.Further every point of R is a limit point, for every nbd of any of its points contains an infinite member of R.
Most Upvoted Answer
Which one of the following is not correct?a)The set I has no limit poi...
The limits points of Q is R. every points of Q as well as every points of R-Q is limits points of Q.
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Community Answer
Which one of the following is not correct?a)The set I has no limit poi...
Explanation:
To understand why option C is the correct answer, let's define some terms and discuss each option one by one.
Limit Point:
A point p is said to be a limit point of a set S if every open neighborhood of p contains at least one point of S other than p itself. In other words, for any open ball centered at p, there exists a point in the set S within that ball.
Option A: The set I has no limit point.
This option is incorrect. The set of real numbers between 0 and 1, denoted by I, does have limit points. For example, if we consider the interval (0, 1), then every point in the interval is a limit point, as there are infinitely many points in I within any open neighborhood around the point.
Option B: The set hi has no limit point.
It is unclear what "hi" refers to in this option. Without further information, we cannot determine whether this option is correct or not.
Option C: No point of Q is a limit point.
This option is correct. The set Q represents the set of rational numbers. A rational number can be expressed as a fraction of two integers. The rational numbers are dense in the real number line, meaning that between any two rational numbers, there exists another rational number. As a result, there is always another rational number within any open neighborhood of a rational number. Therefore, no point in Q can be a limit point.
Option D: Every point of the set M is a limit point.
Without information about the set M, we cannot determine whether this option is correct or not.
Therefore, the correct answer is option C, as explained above.
To understand why option C is the correct answer, let's define some terms and discuss each option one by one.
Limit Point:
A point p is said to be a limit point of a set S if every open neighborhood of p contains at least one point of S other than p itself. In other words, for any open ball centered at p, there exists a point in the set S within that ball.
Option A: The set I has no limit point.
This option is incorrect. The set of real numbers between 0 and 1, denoted by I, does have limit points. For example, if we consider the interval (0, 1), then every point in the interval is a limit point, as there are infinitely many points in I within any open neighborhood around the point.
Option B: The set hi has no limit point.
It is unclear what "hi" refers to in this option. Without further information, we cannot determine whether this option is correct or not.
Option C: No point of Q is a limit point.
This option is correct. The set Q represents the set of rational numbers. A rational number can be expressed as a fraction of two integers. The rational numbers are dense in the real number line, meaning that between any two rational numbers, there exists another rational number. As a result, there is always another rational number within any open neighborhood of a rational number. Therefore, no point in Q can be a limit point.
Option D: Every point of the set M is a limit point.
Without information about the set M, we cannot determine whether this option is correct or not.
Therefore, the correct answer is option C, as explained above.
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Which one of the following is not correct?a)The set I has no limit point.b)The set hi has no limit point.c)No point of Q is a limit point.d)Every point of the set M is a limit point.Correct answer is option 'C'. Can you explain this answer?
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