Let P ( S ) denote power set of S ={1,2,3, ldots, 10} . Define relations R₁ and R₂ on P(S) as A

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1.Relation and Function

hard

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 $A R_1 B$ if $\left( A \cap B ^{ c }\right) \cup\left( B \cap A ^{ c }\right)=\varnothing$ and $AR _2 B$ if $A \cup B ^{ c }=$ $B \cup A ^{ c }, \forall A , B \in P ( S )$. Then :

both $R_1$ and $R_2$ are equivalence relations

only $R_1$ is an equivalence relation

only $R_2$ is an equivalence relation

both $R_1$ and $R_2$ are not equivalence relations

(JEE MAIN-2023)

Solution

$S=\{1,2,3, \ldots \ldots .10\}$

$P ( S )=$ power set of $S$

$AR , B \Rightarrow( A \cap \overrightarrow{ B }) \cup(\overrightarrow{ A } \cap B )=\phi$

$R 1$ is reflexive, symmetric

For transitive

$( A \cap \overrightarrow{ B }) \cup(\overrightarrow{ A } \cap B )=\phi ;\{ a \}=\phi=\{ b \} A = B$

$( B \cap \overrightarrow{ C }) \cup(\overrightarrow{ B } \cap C )=\phi \therefore B = C \therefore A = C \text { equivalence. }$

$R _2 \equiv A \cup \overrightarrow{ B }=\overrightarrow{ A } \cup B$

$R _2 \rightarrow \text { Reflexive, symmetric }$

for transitive

$\begin{array}{l} A \cup \overrightarrow{ B }=\overrightarrow{ A } \cup B \Rightarrow\{ a , c , d \}=\{ b , c , d \} \\ \{ a \}=\{ b \} \therefore A = B \\ B \cup \overrightarrow{ C }=\overrightarrow{ B } \cup C \Rightarrow B = C \\ \therefore A = C \quad \therefore A \cup \overrightarrow{ C }=\overrightarrow{ A } \cup C \therefore \text { Equivalence }\end{array}$

Standard 12

Mathematics

Let $A=\{1,2,3,4\}$ and $R$ be a relation on the set $A \times A$ defined by $R=\{((a, b),(c, d)): 2 a+3 b=4 c+5 d\}$. Then the number of elements in $R$ is:

normal

(JEE MAIN-2023)

Let $A =\{1,2,3, \ldots, 10\}$ and $R$ be a relation on $A$ such that $R =\{( a , b ): a =2 b+1\}$. Let $\left( a _1, a _2\right)$, $\left(a_2, a_3\right),\left(a_3, a_4\right), \ldots .,\left(a_k, a_{k+1}\right)$ be a sequence of $k$ elements of $R$ such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer $k$ , for which such a sequence exists, is equal to :

normal

(JEE MAIN-2025)

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 $A R_1 B$ if $\left( A \cap B ^{ c }\right) \cup\left( B \cap A ^{ c }\right)=\varnothing$ and $AR _2 B$ if $A \cup B ^{ c }=$ $B \cup A ^{ c }, \forall A , B \in P ( S )$. Then :

Solution

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