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 AR_1B if (Acap B^c )cup (Bcap A^c )=phi and AR_2B if Acup B^c=Bcup A^c , forall A,Bin P(S) . Then :

Sets, Relations and Function: Let P(S) denote the power set of S=\1,2,3,ldots,10\ . Define the… | JEE Main 2023

JEE Main 2023 (Online) 1st February Evening ShiftQuestion 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 $AR_1B$ if $\left(A\cap B^{c}\right)\cup\left(B\cap A^{c}\right)=\phi$ and $AR_2B$ if $A\cup B^{c}=B\cup A^{c}$, $\forall A,B\in P(S)$. Then :

Sets, Relations and Function

both $R_1$ and $R_2$ are equivalence relations✓ Correct answeronly $R_1$ is an equivalence relationonly $R_2$ is an equivalence relationboth $R_1$ and $R_2$ are not equivalence relations

Solution

$(A\cap B^c)\cup(B\cap A^c)=\phi$ means $A\triangle B=\phi$, i.e. $A=B$. $A\cup B^c=B\cup A^c$ fails at any $x\in A\setminus B$ (left side contains $x$, right side does not), and similarly for $B\setminus A$, so again $A=B$. Both are the equality relation, so both are equivalence relations.

Sets, Relations and Function - JEE Mathematics question (JEE Main 2023 (Online) 1st February Evening Shift)

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