1
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1

Let $$P(S)$$ denote the power set of $$S=\{1,2,3, \ldots ., 10\}$$. Define the relations $$R_{1}$$ and $$R_{2}$$ on $$P(S)$$ as $$\mathrm{AR}_{1} \mathrm{~B}$$ if $$\left(\mathrm{A} \cap \mathrm{B}^{\mathrm{c}}\right) \cup\left(\mathrm{B} \cap \mathrm{A}^{\mathrm{c}}\right)=\emptyset$$ and $$\mathrm{AR}_{2} \mathrm{~B}$$ if $$\mathrm{A} \cup \mathrm{B}^{\mathrm{c}}=\mathrm{B} \cup \mathrm{A}^{\mathrm{c}}, \forall \mathrm{A}, \mathrm{B} \in \mathrm{P}(\mathrm{S})$$. Then :

A
only $$R_{2}$$ is an equivalence relation
B
both $$R_{1}$$ and $$R_{2}$$ are not equivalence relations
C
both $$R_{1}$$ and $$R_{2}$$ are equivalence relations
D
only $$R_{1}$$ is an equivalence relation
2
JEE Main 2023 (Online) 1st February Morning Shift
+4
-1

Let $$R$$ be a relation on $$1\mathbb{R}$$, given by $$R=\{(a, b): 3 a-3 b+\sqrt{7}$$ is an irrational number $$\}$$. Then $$R$$ is

A
an equivalence relation
B
reflexive and symmetric but not transitive
C
reflexive and transitive but not symmetric
D
reflexive but neither symmetric nor transitive
3
JEE Main 2023 (Online) 31st January Evening Shift
+4
-1
Among the relations

$\mathrm{S}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathbb{R}-\{0\}, 2+\frac{\mathrm{a}}{\mathrm{b}}>0\right\}$

and $\mathrm{T}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathbb{R}, \mathrm{a}^{2}-\mathrm{b}^{2} \in \mathbb{Z}\right\}$,
A
$\mathrm{S}$ is transitive but $\mathrm{T}$ is not
B
both $\mathrm{S}$ and $\mathrm{T}$ are symmetric
C
neither $S$ nor $T$ is transitive
D
$T$ is symmetric but $S$ is not
4
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1

Let $$\mathrm{R}$$ be a relation on $$\mathrm{N} \times \mathbb{N}$$ defined by $$(a, b) ~\mathrm{R}~(c, d)$$ if and only if $$a d(b-c)=b c(a-d)$$. Then $$\mathrm{R}$$ is

A
symmetric and transitive but not reflexive
B
reflexive and symmetric but not transitive
C
transitive but neither reflexive nor symmetric
D
symmetric but neither reflexive nor transitive
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