AIEEE 2011 (Offline)Question 1
Let $R$ be the set of real numbers.
Statement I: $A=\{(x,y)\in R\times R:y-x\text{ is an integer}\}$ is an equivalence relation on $R$.
Statement II: $B=\{(x,y)\in R\times R:x=\alpha y\text{ for some rational number }\alpha\}$ is an equivalence relation on $R$.
Sets, Relations and Function
Solution
$y-x\in\mathbb Z$ is reflexive, symmetric and transitive, so $A$ is an equivalence relation. For $B$: $(0,1)\in B$ with $\alpha=0$, but $1=\alpha\cdot0$ is impossible, so $B$ is not symmetric. Statement I true, Statement II false.