Question

9. Show that set of all rational numbers is not a closed set and also it is not an open set.​

Answer 1

Answer:

The set of all rational numbers is neither a closed set nor an open set.

Rational numbers are numbers that are made by dividing two integers and can be expressed as a quotient or fraction.

An open set is a group of numbers that don't have any boundaries. To understand it better, it can be observed in your standard number line where an open circle on a number means that it is not included in the set. Therefore, the set is an open set since it does not have a definite limit. On the other hand, closed sets are determined to be the complement of open sets where the boundaries are defined which is represented by a solid circle on your number line.

The set of rational numbers is determined to be neither an open set nor a closed set:

° The set of rational numbers is not considered open since each neighborhood of the numbers in the set holds an irrational number.

° Also, the complement is also not considered open since each neighborhood of the irrational numbers as a rational number.

Conclusion:

Hence, the set of all rational numbers is not closed and also it is not an open set.

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Answer 2

Answer:

• When two integers are divided, a number is created that is rational and can be stated as a quotient or fraction.
• A collection of numbers with no boundaries is referred to as an open set. It may be seen in your typical number line, where an open circle on a number indicates that it is not part of the set, to better understand it.
• As a result, the set is an open set because it lacks a specified limit. On the other hand, closed sets, where the boundaries are known and are denoted by a solid circle on your number line, are found to be the complement of open sets.
• It is established that the set of rational numbers is neither an open set nor a closed set.
• The set of rational numbers is not regarded as open because each of its neighborhoods contains an irrational number.
• Additionally, because each neighborhood of the irrational numbers is treated as a rational number, the complement is also not regarded as open.

Conclusion:

As a result, neither the set of all rational numbers is closed nor is it an open set for the collection of all rational numbers.

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