Let S be the set of all rational numbers in (0, 1). Then which of the following statements is/are TRUE ?a)S is a closed subset of Rb)S is not a closed subset of Rc)S is an open subset of Rd)Every x (0, 1)\S is a limit point of S.Correct answer is option 'B,D'. Can you explain this answer?

Question

Answer

Solution:

Statement a) S is a closed subset of R:
To determine if S is a closed subset of R, we need to consider its complement, which is the set of all irrational numbers and the numbers outside the interval (0, 1).

The set of irrational numbers is not closed since it contains numbers like √2 that are limit points but not contained within the set. Similarly, the set of numbers outside the interval (0, 1) is not closed since it contains numbers like 1 and 0 that are also limit points but not contained in the set.

Therefore, the complement of S is not closed, which implies that S itself is not closed. Hence, statement a) is false.

Statement b) S is not a closed subset of R:
As explained above, the complement of S, which includes the set of irrational numbers and the numbers outside the interval (0, 1), is not closed. This implies that S cannot be closed since the definition of a closed set is that its complement is closed. Therefore, statement b) is true.

Statement c) S is an open subset of R:
To determine if S is an open subset of R, we need to check if every point in S has a neighborhood contained within S.

Since S consists of all rational numbers in the interval (0, 1), any neighborhood around a rational number in S will contain both rational and irrational numbers. Therefore, it is not possible to find a neighborhood that is entirely contained within S. Hence, statement c) is false.

Statement d) Every x (0, 1)\S is a limit point of S:
To determine if every x in (0, 1)\S is a limit point of S, we need to consider the complement of S and check if every point in it is a limit point of S.

The complement of S consists of irrational numbers and the numbers outside the interval (0, 1). Any irrational number in the complement is a limit point of S since there are infinitely many rational numbers arbitrarily close to it. Similarly, any number outside the interval (0, 1) is also a limit point of S since there are rational numbers in S arbitrarily close to it. Therefore, every point in the complement of S is a limit point of S. Hence, statement d) is true.

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