Q191 / 270 · NDA 2026 (Phase 1) · Mathematics
Let $R$ be a relation on the set $N$ of natural numbers defined by $R=\{(x,y):x,y\in N \text{ and } x=y^3\}$. Which of the following statements is/are correct ?
I. $R$ is symmetric relation
II. $R$ is transitive relation
Select the answer using the code given below :
I. $R$ is symmetric relation
II. $R$ is transitive relation
Select the answer using the code given below :
- AI only
- BII only
- CBoth I and II
- Correct answerDNeither I nor II
Solution
Given relation $R = \{(x,y) : x, y \in N \text{ and } x = y^3\}$.
For symmetry:
Let $x = 8$ and $y = 2$.
Since $8 = 2^3$, $(8, 2) \in R$.
However, $2 \neq 8^3$, so $(2, 8) \notin R$.
Thus, $R$ is not symmetric. Statement I is incorrect.
For transitivity:
Let $x = 512$, $y = 8$, and $z = 2$.
Since $512 = 8^3$, $(512, 8) \in R$.
Since $8 = 2^3$, $(8, 2) \in R$.
However, $512 \neq 2^3$, so $(512, 2) \notin R$.
Thus, $R$ is not transitive. Statement II is incorrect.
Neither I nor II is correct.
Answer: Neither I nor II
For symmetry:
Let $x = 8$ and $y = 2$.
Since $8 = 2^3$, $(8, 2) \in R$.
However, $2 \neq 8^3$, so $(2, 8) \notin R$.
Thus, $R$ is not symmetric. Statement I is incorrect.
For transitivity:
Let $x = 512$, $y = 8$, and $z = 2$.
Since $512 = 8^3$, $(512, 8) \in R$.
Since $8 = 2^3$, $(8, 2) \in R$.
However, $512 \neq 2^3$, so $(512, 2) \notin R$.
Thus, $R$ is not transitive. Statement II is incorrect.
Neither I nor II is correct.
Answer: Neither I nor II
Hence, the correct answer is option: 'D'.