Practicing Success
For any real numbers x and y, we define x R y if and only if $cosec^2x-cot^2y=1$. The relation R is : |
Reflexive but not transitive Symmetric but not reflexive both reflexive and symmetric but not transitive an equivalence relation |
an equivalence relation |
The correct answer is Option (4) → an equivalence relation $cosec^2x=1+\cot^2y$ $⇒cosec^2y=cosec^2x$ $|cosec^2y|=|cosec^2x|$ for all x ∈ domain of $cosec\, x$ $|cosec\, x|=|cosec\, x|$ ⇒ Reflexive for every $(x, y)∈R$ $|cosec\, x|=|cosec\, y|⇒(y,x)∈R$ ⇒ Symmetric for $(x,y)∈R,(y,z)∈R$ $|cosec\, x|=|cosec\, y|, |cosec\, y|=|cosec\, z|$ $⇒|cosec\, x|=|cosec\, z|⇒(x,z)∈R$ ⇒ Transitive ⇒ R is equivalence relation |