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Let A be a nonempty subset of R Let I(A) denote the set of interior points of A. Then I(A) can be
- a)empty
- b)singleton
- c)a finite set containing more than one element
- d)countable but not finite
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Let A be a nonempty subset of R Let I(A) denote the set of interior po...
Explanation:
To understand why option 'A' is the correct answer, let's first define what interior points are.
Definition: A point x is said to be an interior point of a set A if there exists an open interval containing x that is entirely contained in A.
Now, let's consider the given nonempty subset A of the real numbers.
Case 1: A has no interior points
If A has no interior points, it means that for every point x in A, there is no open interval containing x that is entirely contained in A. In other words, every point in A is either on the boundary of A or outside of A. In this case, the set I(A) of interior points will be empty.
Case 2: A has at least one interior point
If A has at least one interior point, it means that there exists a point x in A such that there is an open interval containing x that is entirely contained in A. In this case, the set I(A) of interior points will contain at least one element.
Since the question states that A is a nonempty subset of R, it implies that A must have at least one element. Therefore, it is possible for A to have interior points, and thus option 'A' is correct.
Summary:
In summary, the set of interior points I(A) can be empty if A has no interior points. However, if A has at least one interior point, then the set I(A) will contain at least one element. Therefore, the correct answer is option 'A'.
To understand why option 'A' is the correct answer, let's first define what interior points are.
Definition: A point x is said to be an interior point of a set A if there exists an open interval containing x that is entirely contained in A.
Now, let's consider the given nonempty subset A of the real numbers.
Case 1: A has no interior points
If A has no interior points, it means that for every point x in A, there is no open interval containing x that is entirely contained in A. In other words, every point in A is either on the boundary of A or outside of A. In this case, the set I(A) of interior points will be empty.
Case 2: A has at least one interior point
If A has at least one interior point, it means that there exists a point x in A such that there is an open interval containing x that is entirely contained in A. In this case, the set I(A) of interior points will contain at least one element.
Since the question states that A is a nonempty subset of R, it implies that A must have at least one element. Therefore, it is possible for A to have interior points, and thus option 'A' is correct.
Summary:
In summary, the set of interior points I(A) can be empty if A has no interior points. However, if A has at least one interior point, then the set I(A) will contain at least one element. Therefore, the correct answer is option 'A'.
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Community Answer
Let A be a nonempty subset of R Let I(A) denote the set of interior po...
Empty, Since if a set contains an interior point, infact it contains infinitely many interior points
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Let A be a nonempty subset of R Let I(A) denote the set of interior points of A. Then I(A) can bea)emptyb)singletonc)a finite set containing more than one elementd)countable but not finiteCorrect answer is option 'A'. Can you explain this answer?
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