JEE Main 2024 (Online) 8th April Evening ShiftQuestion 1
Let $A=\{2,3,6,8,9,11\}$ and $B=\{1,4,5,10,15\}$. Let $R$ be a relation on $A\times B$ defined by $(a,b)R(c,d)$ if and only if $3ad-7bc$ is an even integer. Then the relation $R$ is
Sets, Relations and Function
Solution
$3ad-7bc\equiv ad+bc \pmod 2$, so $(a,b)R(c,d)\iff ad+bc$ even. Reflexive: $(a,b)R(a,b)$ needs $2ab$ even: true. Symmetric: $ad+bc$ is symmetric in swapping $(a,b)\leftrightarrow(c,d)$: true. Transitive fails: $(2,1)R(2,4)$ since $2\cdot4+1\cdot2=10$ is even; $(2,4)R(3,1)$ since $2\cdot1+4\cdot3=14$ is even; but $(2,1)R(3,1)$ gives $2\cdot1+1\cdot3=5$, odd. So reflexive and symmetric but not transitive: option D.