Question

Which of the following statements are true? A. Set \(A=\{x:x\in R\text{ and }2<x<3\}\) is a null set. B. Set \(A=\{x:x\in R\text{ and }2<x<4\}\) is a singleton set. C. Set \(A=\{x:x\in R\text{ and }1<x<9\}\) is an infinite set. D. Set \(A=\{x:x\in R\text{ and }1<x<9\}\) is a finite set. E. Set \(A=\{a,b,c,d,e\}\) and \(B=\{c,d,a,e,b\}\) are equal sets.

• A. A and C only
• B. B and D only
• C. D and C only
• D. C and E only

Hint:

For sets of real numbers over an interval, remember that every open interval contains infinitely many real numbers.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks us to identify the correct statements among different classifications of sets (null, singleton, finite, infinite, and equal sets).

Step 2: Detailed Explanation:
Let us analyze the given sets carefully. We must evaluate them based on the standard domain specified in the question:
• Analyzing under Real Numbers ($\mathbf{R$):}
• A: There are infinitely many real numbers between 2 and 3 (e.g., 2.1, 2.01, 2.5). Thus, the set is not a null set (empty set). This statement is false.

• B: There are infinitely many real numbers between 2 and 4. It is not a singleton set (a set containing exactly one element). Thus, this statement is false.

• C: There are infinitely many real numbers between 1 and 9. Thus, it is an infinite set. This statement is true.

• D: Since the set is infinite, it cannot be finite. This statement is false.

• E: Two sets are equal if they contain exactly the same elements. The order of elements does not matter. Both set A and set B have the elements $\{a, b, c, d, e\}$. Thus, they are equal sets. This statement is true.
If $\mathbf{R}$ is strictly used, then statements C and E are correct. However, "C and E" is not available in the given options.
• Analyzing under Integers ($\mathbf{Z$) (Possible Typo in Exam Question):}
Often in competitive exams, $\mathbf{R}$ is printed by mistake instead of $\mathbf{Z}$ (Integers) or $\mathbf{N}$ (Natural Numbers). Let us evaluate if $x \in \mathbf{Z}$:
• A: There are no integers strictly between 2 and 3. Thus, the set is empty (null set). (True)

• B: The only integer strictly between 2 and 4 is 3. Thus, the set is $\{3\}$, which is a singleton set. (True)

• C: The integers between 1 and 9 are $\{2, 3, 4, 5, 6, 7, 8\}$. This is a finite set, not infinite. (False)

• D: Since the set of integers is finite, this statement would be true.

• E: Set E is always true because the elements are identical. (True)
If the domain was intended to be integers, then statements A, B, D, and E are true. Looking at the options, option (D) is "B and E only", which are both true statements.

Step 3: Final Answer:
Under the most consistent set interpretation, statements B and E are selected, making option (D) the correct choice.