A. Set A = {x : x $\in$ R and 2 < x < 3} is a null set.
B. Set A = {x : x $\in$ R and 2 < x < 4} is a singleton set.
C. Set A = {x : x $\in$ R and 1 < x < 9} is an infinite set.
D. Set A = {x : x $\in$ R 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.
Choose the correct answer from the options given below :
Set Analysis: Identifying True Statements
The question asks to identify which statements about sets are true. Let's analyze each statement:
Statement A Analysis
Set A = {x : x $\in$ R and 2 < x < 3} is described as a null set.
- A null set (or empty set) contains no elements.
- The set A contains all real numbers strictly between 2 and 3.
- Examples include 2.1, 2.5, $\sqrt{5}$, etc.
- Since there are infinitely many real numbers between 2 and 3, the set is not empty.
- Therefore, Statement A is false.
Statement B Analysis
Set A = {x : x $\in$ R and 2 < x < 4} is described as a singleton set.
- A singleton set contains exactly one element.
- The set A contains all real numbers strictly between 2 and 4.
- Examples include 2.5, 3, $\sqrt{10}$, etc.
- Since there are infinitely many real numbers between 2 and 4, the set is not a singleton.
- Therefore, Statement B is false.
Statement C Analysis
Set A = {x : x $\in$ R and 1 < x < 9} is described as an infinite set.
- The set A contains all real numbers strictly between 1 and 9.
- There are infinitely many real numbers in this interval (e.g., 1.1, 5, $\pi$, 8.99).
- Therefore, Statement C is true.
Statement D Analysis
Set A = {x : x $\in$ R and 1 < x < 9} is described as a finite set.
- As established in the analysis of Statement C, this set contains infinitely many elements.
- A finite set contains a countable number of elements.
- Therefore, Statement D is false.
Statement E Analysis
Set A = {a, b, c, d, e} and Set B = {c, d, a, e, b} are described as equal sets.
- Two sets are considered equal if and only if they contain exactly the same elements, regardless of the order of elements.
- Set A contains {a, b, c, d, e}.
- Set B contains {c, d, a, e, b}.
- Both sets contain the same five distinct elements.
- Therefore, Statement E is true.
Conclusion
The true statements are C and E.
The correct option is the one that lists both C and E.
Correct Option: 4. C and E only
Important Questions from Set Theory and Venn Diagram (Notes)
A student is free to choose only Chemistry, only Biology or both. If out of $32$ students, Chemistry has been chosen by $16$ and Biology by $25$, then how many students have chosen Biology but not Chemistry?
- Fourteen of the students in a class are girls. Eight students in the class wear blue shirts. Two are neither girls nor wear blue shirts. Five students who wear blue shirts are girls. How many students are there in the class?
- In a group of 44 players, 26 play hockey, 24 play football and 24 play cricket. Eight of them play both hockey and football, 12 play both football and cricket, and 5 play all the three games. How many play both hockey and cricket?
- In a group of students, 30% play only cricket, 20% play only football and 10% play only basketball. 20% of the students play both football and cricket, 15% play both basketball and cricket, 10% play both football and basketball. 15 students play no games, while 5% of the students play all three games. What is the total number of students?
- Suppose three meetings of a group of professors were arranged in Mumbai, Delhi and Chennai. Each professor of the group attended exactly two meetings. 21 professors attended Mumbai meeting, 27 attended Delhi meeting and 30 attended Chennai meeting. How many of them attended both the Chennai and Delhi meetings?
