JEE Main202529 Jan 2025Morning ShiftMathematicsSets and RelationsActual
Define a relation R on the interval ( [0, 2 ) ) by (x R y ) if and only if ( ^2 x- ^2 y=1 ). Then R is :
Options
- Aboth reflexive and transitive but not symmetric
- Ban equivalence relation
- Creflexive but neither symmetric not transitive
- Dboth reflexive and symmetric but not transitive
Correct answer
B. an equivalence relation
Step-by-step solution
aligned & ^2 x- ^2 x=1 ( on replacing y with x) & Reflexive & ^2 x- ^2 y=1 & 1+ ^2 x+1- ^2 y=1 & ^2 y- ^2 x=1 & symmetric & ^2 x- ^2 y=1 & ^2 y- ^2 z=1 aligned Adding both aligned & ^2 x- ^2 y+ ^2 y- ^2 z=1+1 & ^2 x+1- ^2 z=2 & ^2 x- ^2 z=1 & Transitive aligned hence equivalence releation