Class JEE Mathematics Sets, Relations, and Functions Q #1056
KNOWLEDGE BASED
APPLY
4 Marks 2020 JEE Main 2020 (Online) 3rd September Evening Slot MCQ SINGLE
Let $R_1$ and $R_2$ be two relation defined as follows: $R_1 = {(a, b) \in R^2 : a^2 + b^2 \in Q}$ and $R_2 = {(a, b) \in R^2 : a^2 + b^2 \notin Q}$, where $Q$ is the set of all rational numbers. Then :
(A) Neither $R_1$ nor $R_2$ is transitive.
(B) $R_2$ is transitive but $R_1$ is not transitive.
(C) $R_1$ and $R_2$ are both transitive.
(D) $R_1$ is transitive but $R_2$ is not transitive.
Prev Next
Correct Answer: A
Explanation
For $R_1$: Let $a = 1 + \sqrt{2}$, $b = 1 - \sqrt{2}$, $c = \sqrt[4]{8}$. $aR_1b : a^2 + b^2 = 6 \in Q$ $bR_1c : b^2 + c^2 = 3 - 2\sqrt{2} + 2\sqrt{2} = 3 \in Q$ $aR_1c : a^2 + c^2 = 3 + 2\sqrt{2} + 2\sqrt{2} \notin Q$ $\therefore$ $R_1$ is not transitive. For $R_2$: Let $a = 1 + \sqrt{2}$, $b = \sqrt{2}$, $c = 1 - \sqrt{2}$ $aR_2b : a^2 + b^2 = 5 + 2\sqrt{2} \notin Q$ $bR_2c : b^2 + c^2 = 5 - 2\sqrt{2} \notin Q$ $aR_2c : a^2 + c^2 = 3 + 2\sqrt{2} + 3 - 2\sqrt{2} = 6 \in Q$ $\therefore$ $R_2$ is not transitive.

More from this Chapter

NUMERICAL
Let $S=\{4,6,9\}$ and $T=\{9,10,11, \ldots, 1000\}$. If $A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in \mathbf{N}, a_{1}, a_{2}, a_{3}, \ldots, a_{k}\right.$ $\epsilon S\}$, then the sum of all the elements in the set $T-A$ is equal to __________.
NUMERICAL
Let $$A = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\min \,\{ i,j\} } } $$ and $$B = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\max \,\{ i,j\} } } $$Then A + B is equal to _____________.
MCQ_SINGLE
Let $R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\}$ be a relation on the set $A = \{1, 2, 3, 4\}$. The relation $R$ is:
NUMERICAL
Let $A=\{1,2,3, \ldots, 20\}$. Let $R_1$ and $R_2$ two relation on $A$ such that $R_1=\{(a, b): b$ is divisible by $a\}$ $R_2=\{(a, b): a$ is an integral multiple of $b\}$. Then, number of elements in $R_1-R_2$ is equal to _____________.
MCQ_SINGLE
Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as $R_1 = \{(x, y) \in N \times N : 2x + y = 10\}$ and $R_2 = \{(x, y) \in N \times N : x + 2y = 10\}$. Then :
View All Questions