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

Check statement A. \[ A=\{x:x\in R\text{ and }2<x<3\}. \] There are infinitely many real numbers between \(2\) and \(3\). For example: \[ 2.1,\ 2.2,\ 2.5,\ 2.75 \] all belong to this set. So this is not a null set. Therefore, A is incorrect. Check statement B. \[ A=\{x:x\in R\text{ and }2<x<4\}. \] There are infinitely many real numbers between \(2\) and \(4\). So this is not a singleton set. Therefore, B is incorrect. Check statement C. \[ A=\{x:x\in R\text{ and }1<x<9\}. \] There are infinitely many real numbers between \(1\) and \(9\). Therefore, C is correct. Check statement D. The same set: \[ \{x:x\in R\text{ and }1<x<9\} \] is not finite because it contains infinitely many real numbers. Therefore, D is incorrect. Check statement E. \[ A=\{a,b,c,d,e\} \] and: \[ B=\{c,d,a,e,b\}. \] Both sets contain exactly the same elements. Order does not matter in sets. Therefore: \[ A=B. \] So E is correct. Thus, mathematically correct statements are: \[ C\text{ and }E. \]