Let R be the set of real numbers This question has Statement -1 and Statement -2 . Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1 : A= (x, y) R R: y-x is an integer is an equivalence relation on R . Statement-2 : B= (x, y) R R: x= y for some rational number is an equivalence relation on R .

Step-by-step solution

x - y is an integer x-x=0 is an integer A is Reflexive x-y is an integer y-x is an integer A is symmetric x-y, y-z are integers As sum of two integers is an integer. (x-y)+(y-z)=x-z is an integer A is transitive. Hence statement -1 is true. Also x x =1 is a rational number B is reflexive x y = is rational y x need not be rational i.e., 0 1 is rational 1 0 is not rational Hence B is not symmetric B is not an equivalence relation.