JEE Main 2022 (Online) 29th June Morning ShiftQuestion 1
Let a set $A=A_1\cup A_2\cup\ldots\cup A_k$, where $A_i\cap A_j=\phi$ for $i\neq j$, $1\le i,j\le k$. Define the relation $R$ from $A$ to $A$ by $R=\{(x,y):y\in A_i$ if and only if $x\in A_i,\ 1\le i\le k\}$. Then, $R$ is :
Sets, Relations and Function
Solution
Reflexive: $x\in A_i\iff x\in A_i$. Symmetric: the condition is symmetric in $x,y$. Transitive: $xRy,yRz$ with $x\in A_i$ gives $y\in A_i$, then $z\in A_i$, so $xRz$. $R$ is an equivalence relation: option D.