JEE Main · 2011 · 10 Apr 2011 · Offline · Sets Relations and Functions

For $A$, $x-x=0$ is an integer so it is reflexive; if $y-x$ is an integer so is $x-y$, giving symmetry; and $(y-x)+(z-y)=z-x$ is an integer, giving transitivity, so $A$ is an equivalence relation and the first statement is true. For $B$, $x=1\cdot x$ makes it reflexive, but it fails symmetry: $0=0\cdot y$ holds for every $y$, yet $y=\alpha \cdot 0$ forces $y=0$, so a related pair $(0,y)$ does not give $(y,0)$ related. Thus $B$ is not an equivalence relation and the second statement is false.