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?
- 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?
Match List-I with List-II
List-1 List-II (A) If X and Y are two sets such that n(X)= 17, n(Y)=23, n(X $\cup$ Y)=38, then n(X $\cap$ Y) is (I) 20 (B)) If n(X) = 28,n(Y) = 32,n(X$\cap$Y) = 10, then n(X$\cup$Y) is (II) 10 (C) If n(X) = 10, then n(7X) is (III) 50 (D) If n(Y) = 20, then n($\frac{Y}{2}$) is (IV) 2 Choose the Correct answer from the options given below:
Consider the following relation R={(4,5),(5,4), (7,6),(6,7)} on set I={4,5,6,7}. Which of the following properties relation R does not have?
A. Reflexive property
B. Symmetric property
C. Transitive property
D. Antisymmetric property
Choose the correct answer from the options given below:Find the least upper bound and greatest lower bound of $S=\{X,Y,Z\}$ if they exist, of the poset whose Hasse diagram is shown below:
