1
JEE Main 2023 (Online) 10th April Evening Shift
+4
-1

Let $$\mathrm{A}=\{2,3,4\}$$ and $$\mathrm{B}=\{8,9,12\}$$. Then the number of elements in the relation $$\mathrm{R}=\left\{\left(\left(a_{1}, \mathrm{~b}_{1}\right),\left(a_{2}, \mathrm{~b}_{2}\right)\right) \in(A \times B, A \times B): a_{1}\right.$$ divides $$\mathrm{b}_{2}$$ and $$\mathrm{a}_{2}$$ divides $$\left.\mathrm{b}_{1}\right\}$$ is :

A
18
B
24
C
36
D
12
2
JEE Main 2023 (Online) 8th April Evening Shift
+4
-1

Let $$\mathrm{A}=\{1,2,3,4,5,6,7\}$$. Then the relation $$\mathrm{R}=\{(x, y) \in \mathrm{A} \times \mathrm{A}: x+y=7\}$$ is :

A
reflexive but neither symmetric nor transitive
B
transitive but neither symmetric nor reflexive
C
symmetric but neither reflexive nor transitive
D
an equivalence relation
3
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
4
JEE Main 2023 (Online) 1st February Morning Shift
+4
-1

Let $$R$$ be a relation on $$\mathbb{R}$$, 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
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